![2 Graphs and Groups: Preliminaries
vertex-set is V(G) and whose edge-set is E(G) − F. If S = {u} and F = {e},
we shall, for short, denote G − S and G − F by G − u and G − e, respectively.
If S is a subset of the vertices of G, then G[S] will denote the subgraph of G
induced by S, that is, the subgraph consisting of the vertices in S and all of the
edges joining pairs of vertices from S.
An important modification of the foregoing definition of a graph gives what
is called a directed graph, or digraph for short. In a digraph D = (V(D), A(D))
the set A = A(D) consists of ordered pairs of vertices from V = V(D) and its
elements are called arcs. Again, an arc (u, v) is sometimes denoted by uv when
it is clear from the context whether we are referring to an arc or an edge. The
arc uv is said to be incident to v and incident from u; the vertex u is said to be
adjacent to v whereas v is adjacent from u. The number of arcs incident from a
vertex v is called its out-degree, denoted by degout(v), while the number of arcs
incident to v is called its in-degree and is denoted by degin(v). A digraph is said
to be regular if all of its vertices have the same out-degree or, equivalently, the
same in-degree. Sometimes, when we need to emphasise the fact that a graph
is not directed, we say that it is undirected.
The number of vertices of a graph G or digraph D is called its order and is
generally denoted by n = n(G) or n = n(D), while the number of edges or
arcs is called its size and is denoted by m = m(G) or m = m(D).
A sequence of distinct vertices of a graph, v1, v2, . . . , vk+1, and edges e1, e2,
. . . , ek such that each edge ei = vivi+1 is called a path. If we allow v1 and
vk+1, and only those, to be the same vertex, then we get what is called a
cycle.
The length of a path or a cycle in G is the number of edges in the path or
cycle. A path of length k is denoted by Pk+1 while a cycle of length k is denoted
by Ck. The distance between two vertices u, v in a connected graph G, denoted
by d(u, v), is the length of the shortest path joining u and v. The diameter of G
is the maximum value attained by d(u, v) as u, v run over V(G), and the girth
is the length of the shortest cycle.
In these definitions, if we are dealing with a digraph and the ei = vivi+1
are arcs, then the path or cycle is called a directed path or directed cycle,
respectively.
Given a digraph D, the underlying graph of D is the graph obtained from
D by considering each pair in A(D) to be an unordered pair. Given a graph G,
the digraph
←
→
G is obtained from G by replacing each edge in E(G) by a pair
of oppositely directed arcs. This way, a graph can always be seen as a special
case of a digraph.
We adopt the usual convention of representing graphs and digraphs by draw-
ings in which each vertex is shown by a dot, each edge by a curve joining the](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-22-320.jpg)
![1.2 Groups 3
corresponding pair of dots and each arc (u, v) by a curve with an arrowhead
pointing in the direction from u to v.
A number of definitions on graphs and digraphs will be given as they are
required. However, several standard graph theoretic terms will be used but not
defined in these chapters; these can be found in any of the references [257] or
[259].
1.2 Groups
A permutation group will be a pair (, Y) where Y is a finite set and is a
subgroup of the symmetric group SY, that is, the group of all permutations of
Y. The stabiliser of an element y ∈ Y under the action of is denoted by y
while the orbit of y is denoted by (y). The Orbit-Stabiliser Theorem states
that, for any element y ∈ Y,
|| = |(y)| · |y|.
If the elements of Y are all in one orbit, then (, Y) is said to be a transitive
permutation group and is said to act transitively on Y. The permutation group
is said to act regularly on Y if it acts transitively and the stabiliser of any
element of Y is trivial. By the Orbit-Stabiliser Theorem, this is equivalent to
saying that acts transitively on Y and || = |Y|. Also, acts regularly on Y
is equivalent to saying that, for any y1, y2 ∈ Y, there exists exactly one α ∈
such that α(y1) = y2.
One important regular action of a permutation group arises as follows. Let
be any group, let Y = and, for any α ∈ , let λα be the permutation
of Y defined by λα(β) = αβ. Let L() be the set of all permutations λα for
all α ∈ . Then (L(), Y) defines a permutation group acting regularly on Y.
This is called the left regular representation of the group on itself. One can
similarly consider the right regular representation of the group on itself, and
this is denoted by (R(), Y).
The following is an important generalisation of the previous definitions. If
is a group and H ≤ , let Y = /H be the set of left cosets of H in . For any
α ∈ , let λH
α be a permutation on Y defined by λH
α (βH) = αβH. Let LH()
be the set of all λH
α for all α ∈ . Then (LH(), Y) defines a permutation
group that reduces to the left regular representation of if H = {1}.
Two permutation groups (1, Y1), (2, Y2) are said to be equivalent, denoted
by (1, Y1) ≡ (2, Y2), if there exists a bijective isomorphism φ : 1 → 2
and a bijection f : Y1 → Y2 such that, for all y ∈ Y1 and for all α ∈ 1,
f(α(x)) = φ(α)(f(x)).](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-23-320.jpg)
![1.3 Graphs and groups 7
is the infinite free group on two elements. The dihedral group of degree n is
denoted by Dn. It has order 2n and also has a presentation with two generators:
α, β|α2
= 1, βn
= 1, α−1
βα = β−1
.
Determining a group from a given presentation is not an easy problem. The
reader who doubts this can try to show that the presentations
α, β : αβ2
= β3
α, βα2
= α3
β
and
α, β, γ : α3
= β3
= γ 3
= 1, αγ = γ α−1
, αβα−1
= βγβ−1
both give the trivial group. We shall of course make a very simple use of stan-
dard group presentations where these difficulties do not arise. The book [159]
is a standard reference for advanced work on group presentations.
The reader is referred to [147, 222] for any terms and concepts on group
theory that are used but not defined in these chapters and, in particular, to [49,
62] for more information on permutation groups.
1.3 Graphs and groups
Let G, G be two graphs. A bijection α : V(G) → V(G ) is called an isomor-
phism if
{u, v} ∈ E(G) ⇔ {α(u), α(v)} ∈ E(G ).
The graphs G, G are, in this case, said to be isomorphic, and this is denoted by
G G . Similarly, if D, D are digraphs, then a bijection α : V(D) → V(D ) is
called an isomorphism if
(u, v) ∈ A(D) ⇔ (α(u), α(v)) ∈ A(D ),
and in this case the digraphs D, D are also said to be isomorphic, and again
this is denoted by D D .
If the two graphs, or digraphs, in this definition are the same, then α is said
to be an automorphism of G or of D. The set of automorphisms of a graph or a
digraph is a group under composition of functions, and it is denoted by Aut(G)
or Aut(D).
Note that an automorphism α of G is an element of SV(G), although it is its
induced action on E(G) that determines whether α is an automorphism. This
fact, although clear from the definition of automorphism, is worth emphasising
when beginning to study automorphisms of graphs.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-27-320.jpg)
![10 Graphs and Groups: Preliminaries
Figure 1.6. The eight nonidentical graphs of order 3
Counting nonisomorphic graphs involves consideration of group symme-
tries. For more on this the reader is referred to [103].
1.4 Edge-automorphisms and line-graphs
Although we shall be dealing mostly with Aut(G) and its realisation as the
permutation group (Aut(G), V(G)), let us briefly look at other related groups
associated with G. In this section we shall assume that G is a nontrivial graph,
that is, its edge-set is nonempty.
An edge-automorphism of a graph G is a bijection θ on E(G) such that two
edges e, f are adjacent in G if and only if θ(e), θ(f) are also adjacent in G. The
set of all edge-automorphisms of G is a group under composition of functions,
and it is denoted by Aut1(G).
The concept of edge-automorphisms can perhaps be best understood within
the context of line-graphs. The line-graph L(G) of a graph G is defined as
the graph whose vertex-set is E(G) and in which two vertices are adjacent if
and only if the corresponding edges are adjacent in G. An automorphism of
L(G) is clearly an edge-automorphism of G and (Aut1(G), E(G)) is equivalent
to (Aut(L(G), V(L(G))). In this section we shall give the exact relationship
between Aut1(G) and Aut(G), that is, between the automorphism groups of G
and L(G).
As we described earlier, any automorphism α of G naturally induces a bijec-
tion α̂ on E(G) defined by α̂(uv) = α(u)α(v). It is an important (and easy to](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-30-320.jpg)
![12 Graphs and Groups: Preliminaries
Figure 1.7. Graphs with edge-isomorphisms not induced by isomorphisms
Before proceeding let us first extend the idea of edge-automorphisms on
the edge-set of a graph to that of edge-isomorphisms between edge-sets of
different graphs.
Let G, G be two nontrivial graphs. A bijection θ : E(G) → E(G ) is an
edge-isomorphism if
e, f adjacent in G ⇔ θ(e), θ(f) adjacent in G .
Two graphs are said to be edge-isomorphic if there is an edge-isomorphism
between their edge-sets.
The graphs W1, W2 in Figure 1.7 are edge-isomorphic, although they are
not isomorphic. That is, there is an edge-isomorphism between their edge-
sets that cannot be induced by an isomorphism between their vertex-sets. This
means that their line-graphs are isomorphic even though the two graphs are not
themselves isomorphic.
Also, each of the graphs W3, W4, W5 in the same figure has edge-
automorphisms that are not induced by automorphisms. That is, the group
Aut∗
(Wi) is a strict subgroup of Aut1(Wi). In other words, Aut(L(Wi)) is larger
than Aut(Wi).
The following theorem of Whitney [258] says that these are essentially the
only cases when edge-isomorphisms that are not induced by isomorphisms can
arise. We give the statement of the theorem without proof, which, although
not deep or difficult, would lengthen this introductory chapter without adding
significant new insights.
Theorem 1.3 (Whitney) Let G, G be connected graphs different from the five
graphs in Figure 1.7. Let θ : E(G) → E(G ) be an edge-isomorphism. Then θ
is induced by an isomorphism from G to G .](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-32-320.jpg)
![1.5 A word on issues of computational complexity 13
Whitney’s Theorem and Theorem 1.2 together therefore give the following
corollary.
Corollary 1.4 Let G be a nontrivial graph. Then Aut1(G) = Aut∗(G) if and
only if both of these conditions hold:
(i) not both W1, W2 are components of G;
(ii) none of Wi, i = 3, 4, 5 is components of G.
Moreover, Aut1(G) Aut(G) (that is, Aut(L(G)) Aut(G)) if and only if (i)
and (ii) hold and G has at most one isolated vertex and K2 is not a component
of G.
1.5 A word on issues of computational complexity
Although in this book we shall not concern ourselves with issues of computa-
tional complexity, it is perhaps worthwhile to say a few words in this regard
here in order to put matters into a better perspective. A student reading the def-
initions of isomorphic graphs and automorphisms might think that it is an easy
matter to determine in general whether two given graphs are isomorphic or to
compute the automorphism group of a graph. In fact, this is far from being the
case, and these problems are very hard to crack in practice, at least as far as
present knowledge goes.
In general, one considers that an efficient algorithm exists for finding a solu-
tion to a problem (for example, finding a nontrivial automorphism of a given
graph) if there is a general algorithm such that the number of operations that
it takes to solve the problem is a polynomial function of the size of the input
(say, the number of vertices in the graph); one says that the algorithm solves
the problem in polynomial time.
Of course, several terms in the previous sentence need exact definitions, but
we shall here take an intuitive approach and refer the reader to [36] or [82] for
the exact details on computational complexity.
Those problems for which an efficient (polynomial-time) algorithm exists
form the class denoted by P (which stands for ‘polynomial’). However, there
are several problems for which it is not known whether an efficient algorithm
does exist. In order to tackle this question of computational intractibility, two
important ideas have been developed.
Firstly, the class NP (which stands for ‘nondeterministic polynomial’) is
defined. Roughly (again we refer the reader to the textbooks cited earlier for
the exact details) this class contains all of those problems for which, given a
candidate solution, one can verify in polynomial time that it is in fact a correct](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-33-320.jpg)
![14 Graphs and Groups: Preliminaries
solution. For example, the problem of determining whether a graph has a non-
trivial automorphism is in NP, since, given such a permutation of the vertices,
it is easy to determine in polynomial time that it is an automorphism.
Now the main question in computational complexity is whether P = NP
(clearly P ⊆ NP), and to tackle this question another important idea is intro-
duced. Given two problems A and B, one says that A is (polynomially) reducible
to B if, given an algorithm for solving B, it can be transformed in polynomial
time into an algorithm for solving A. Reducibility therefore introduces a hier-
archy between problems for, if A is reducible to B, then, in a sense, A cannot
be more difficult to solve (computationally) than B. In particular, if there is an
efficient algorithm for solving B, then there is also an efficient algorithm for
solving A.
Now, the question of reducibility took on special significance by the discov-
ery that in the class NP there are problems, called NP-complete, to which any
other problem in NP is reducible. In other words, if an efficient algorithm can
be found for any NP-complete problem, then all problems in NP would have
an efficient algorithm to solve them, and P would be equal to NP.
Now, it is not known whether the problem of determining if two graphs are
isomorphic, which lies clearly in NP, is NP-complete. In fact, if it turns out
that P is not equal to NP, then there is evidence to suggest that the problem of
graph isomorphism might lie strictly between the classes P and NP.
What all this means in practice is that, as far as present knowledge goes, no
general algorithm can determine in a guaranteed reasonable time whether two
graphs are isomorphic, or whether a given graph has a nontrivial automorphism
(these two problems are closely related [126]). It is known that for special types
of graphs (for example, trees, planar graphs and graphs with bounded degree)
an efficient algorithm does exist.
Computer packages can also help one to solve these problems, certainly
more efficiently than an attempt ‘by hand’ for large graphs, although their time
performance is not guaranteed (by what we said earlier). For example, the soft-
ware package MathematicaTM has a combinatorics extension that, amongst
other things, finds graph automorphisms and isomorphisms. A more specialised
package, and one that is freely available from
www.combinatorialmath.org.ca/gg/index.html
is Groups Graphs [131], developed by Bill Kocay. This package contains
several combinatorial routines related to graphs, digraphs, combinatorial designs
and their automorphism groups and also embeddings of graphs on some sur-
faces and a graph isomorphism algorithm. It is easy to use, and it has a pleasant
graphical user interface. It is also very useful simply for drawing diagrams of](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-34-320.jpg)
![1.6 Exercises 15
graphs. Although originally written for MacintoshTM computers, a version for
the unix-based Haiku operating system is in preparation, and this version will
contain several new features.
An important computer algebra package, which is also freely available, is
the system GAP [243]. This package performs very sophisticated routines in
discrete abstract algebra, in particular routines on permutation groups. It incor-
porates a number of extensions, one of which, GRAPE [235], deals specifically
with graphs, including their automorphisms and isomorphisms.
The computer package Sage [227] is an open-source competitor to systems
like MapleTM, MathematicaTM and MatlabTM. It incorporates several open-
source mathematical software like GAP and R, and it can be run via Sage-
MathCloud without the need of installing the system on one’s computer. It has
an excellent library of functions for doing graph theory. In this book we shall
present some constructions using GAP and Sage.
Finally, it should be mentioned that it is generally accepted that the best
package to tackle graph isomorphisms is nauty [181], developed by Brendan
McKay. In fact, the system GRAPE invokes nauty when computing automor-
phisms or isomorphisms.
1.6 Exercises
1.1 Draw all twenty nonidentical graphs with vertex-set {1, 2, 3, 4} that have three
edges. How many of them are nonisomorphic? In general, how many nonidentical
graphs on n vertices and m edges are there? How many are there on n vertices?
1.2 Let G be the graph in Figure 1.8. How many nonidentical labellings does G have
using the labels {1, 2, . . . , 6} on its vertices?
In general, how many nonidentical labellings does a graph G on n vertices
have using the labels {1, 2, . . . , n} on its vertices?
1.3 Show that if G is self-complementary (that is, G G), then n ≡ 0 mod 4 or
n ≡ 1 mod 4. Determine all self-complementary graphs on five vertices.
1.4 A well-known result due to Cayley says that the number of nonidentical trees on
n vertices is nn−2. Verify this for n = 4. Look up one of the several proofs of this
result.
Figure 1.8. How many distinct labellings does this graph have?](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-35-320.jpg)
![16 Graphs and Groups: Preliminaries
The points 1, 2, . . . , n are drawn in a plane. A random tree is drawn joining
these points, with all possible spanning trees being equally likely. Let pn be the
probability that 1 is an endvertex of the tree. Show that limn→∞ pn = 1/e.
1.5 Find a graph with a pair of pseudosimilar edges.
1.6 Let Y = {1, 2, 3, 4} and let be the group acting on Y generated by the permu-
tation (1 2 3 4). Construct a digraph D whose vertex-set is Y and whose arc-set is
the orbit of the arc (1, 2) under the action of the permutation group (, Y × Y).
Is the whole of Aut(D)? Can a graph be obtained by taking the orbit of some
other arc or a union of orbits? Will always be the whole of Aut(D)?
1.7 Let P be a rectangular plate and Q a plate in the form of a rhombus. Show that
the groups of symmetry of P and Q are isomorphic as abstract groups but not
equivalent considered as permutation groups of the four vertices of P and Q.
Show that the same situation arises with the following two graphs: the cycle
on four vertices with an extra multiple edge and the complete graph K4 with an
edge deleted.
1.8 Show that the dihedral group Dn can be presented as
α, β|α2 = β2 = (αβ)n = 1 .
1.9 Let 1 be the abelian group defined by the presentation
α, β, γ |α3 = β3 = γ 3 = 1, [α, β] = [α, γ ] = [β, γ ] = 1 ,
where [α, β] = α−1β−1αβ is the commutator of α and β, and let 2 be the group
defined by the presentation
α, β, γ |α3 = β3 = γ 3 = 1, [β, α] = γ , [α, γ ] = [β, γ ] = 1 .
Show that although 1 and 2 are not isomorphic as abstract groups, and there-
fore the two permutation groups (L(1), 1) and (L(2), 2) are not equivalent,
still the cycle structures of the permutations of L(1) acting on 1 are the same
as those of the permutations of L(2) acting on 2.
Give an example of two permutation groups whose permutations have the
same cycle structures and which are isomorphic as abstract groups but are still
not equivalent as permutation groups.
1.10 Prove Theorem 1.1.
1.11 This exercise is intended to illustrate Theorem 1.1. Consider the action of the
alternating group A4 on the set X = {1, 2, 3, 4}. Let H be the stabiliser of the
element 1 under this action. Confirm that the left action of A4 on the left cosets
of H is equivalent to the action of A4 on X.
1.12 Let be a group of permutations acting on a set Y and let y, x be two elements of
Y that are in the same orbit under this action. Let α, β ∈ be two permutations
such that α(y) = β(y) = x. Prove that αy = βy.
Prove also that if x is in the same orbit as y and γ is a permutation such that
γ (y) = x and γ y = αy, then x = x.
1.13 Show that if the abelian permutation group acts transitively on Y, then its action
on Y is regular.
1.14 (a) Show that a 2-transitive permutation group is primitive.
(b) Show that if Aut(G), for a graph G, is 2-transitive, then either G or its com-
plement is a complete graph.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-36-320.jpg)
![1.7 Notes and guide to references 17
(c) Suppose that the transitive permutation group (, Y), with |Y| finite, is imprim-
itive. Show that the blocks of an imprimitivity block system have equal size.
1.15 Let G be a vertex-transitive graph whose automorphism group acts impri-
mitively on V(G). Show that the subgraphs of G induced by the blocks of an
imprimitivity block system are all isomorphic.
Suppose that each such subgraph is replaced by its complement, leaving the
other edges intact. Let G be the resulting graph. Show that Aut(G ) = Aut(G).
1.16 The Petersen graph can be defined as follows. Let N = {1, 2, 3, 4, 5}, and let the
vertices of the graph be all subsets of N of size 2 in which two vertices are adja-
cent if the corresponding subsets are disjoint. Use this definition and GAP (with
GRAPE) to construct the Petersen graph and to verify some of its properties.
1.7 Notes and guide to references
One of the standard texts on graph theory has, for many years, been [97]. More
recent books that give an excellent coverage of the subject are [28, 61, 257,
259]. The last reference is a short introduction that is quite sufficient back-
ground for this book. Biggs’ book [24] is the standard text on algebraic graph
theory, but the more recent [90] is also an excellent and up-to-date textbook on
the subject. The book [94] contains a number of recent and specialised survey
papers on various aspects of algebraic graph theory, particularly those dealing
with graph symmetries. A proof of Whitney’s Theorem can be found in [22].
We shall need only the most elementary notions of group theory. The text
[147] gives ample coverage for our purposes, while [222] provides a more
complete treatment. Two excellent books devoted entirely to permutation
groups are [49, 62]. Most of the results and definitions on permutation groups
that we have given here and others that we shall need can be found in the first
few chapters of these two books.
For a full discussion of the terms on computational complexity that were
introduced earlier rather intuitively, the reader is referred to the standard text-
book [82] or the more recent [36]. The book [126] and the references it cites
are suggested for those who are interested in the computational complexity of
the graph isomorphism problem. Those who are particularly interested in some
of the powerful algebraic techniques used to tackle this problem should look
at the papers [106, 155]. For practical computations on a computer with per-
mutation groups and graph automorphisms and isomorphisms in particular, the
systems [131, 181, 235, 243] are recommended.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-37-320.jpg)
![20 Various Types of Graph Symmetry
Now let a, b be in the same orbit, say V1. It then follows that a, b are not
adjacent. For suppose otherwise. Then the edge {u, v} is similar to the edge
{a, b} under the action of H; therefore the vertex v is similar to one of a or b
under this action, giving that v (which is in the orbit V2) is also in the orbit V1,
which contains a and b. But this is impossible since V1 ∩ V2 = ∅.
Hence, as required, we have that no two vertices from the same orbit can be
adjacent.
Corollary 2.2 If a graph G without isolated vertices is edge-transitive but not
vertex-transitive, then it is bipartite and the action of Aut(G) on V(G) has two
orbits that form the bipartition of V(G).
Proof Take H = Aut(G) in the previous theorem.
2.1.1 Semisymmetric graphs
The typical example of edge-transitive but not vertex-transitive graphs given
earlier is the complete bipartite graph with a different number of vertices in
the bipartition. These graphs are trivially not vertex-transitive because their
vertices have different degrees. Although regular graphs do exist that are edge-
transitive but not vertex-transitive, it is quite difficult to construct them. Such
graphs are now called semisymmetric graphs and they were first studied by
Folkman [77], who constructed the smallest possible semisymmetric graph
having twenty vertices.1 One construction of the Folkman Graph is described
in Exercise 2.5.
Here we shall describe another well-known semisymmetric graph, the Gray
Graph.2 For a long time nobody could find a smaller cubic semisymmetric
graph, and eventually it was formally proved in [160] that it is the smallest
cubic semisymmetric graph. We shall see a more systematic way of describing
it in subsequent chapters. Here we follow Bouwer’s construction in [33].
Consider a cycle on 54 vertices numbered consecutively from 0 to 53. To
form the Gray Graph G add the following edges to this cycle:
{1, 42}, {2, 15}, {3, 28}, {4, 33}, {5, 44}, {6, 53}, {7, 48}, {8, 21}, {9, 32},
{10, 45}, {11, 24}, {12, 41}, {13, 20}, {14, 31}, {16, 35}, {17, 40}, {18, 49},
{19, 0}, {22, 51}, {23, 30}, {25, 38}, {26, 43}, {27, 52}, {29, 36}, {34, 47},
1 More information about this graph, called the Folkman Graph, can be found on the MathWorld
page http://mathworld. wolfram.com/FolkmanGraph.html
2 The Gray Graph is also featured on MathWorld at http://mathworld.wolfram.com/Gray
Graph.html](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-40-320.jpg)
![2.1 Transitivity 21
{37, 50}, {39, 46}.
It is tedious but not difficult to check that the permutations
α = (2 0 43)(3 53 43)(4 6 44)(7 45 33)(8 10 32)(11 31 21)
= (12 14 20)(15 19 41)(16 18 40)(22 24 30)(25 29 51)
= (26 28 52)(34 48 46)(35 49 39)(36 50 38)
and
β = (1 7 11 37 15 53 9 25 35)(2 6 10 38 16 0 8 24 36)
= (3 5 45 39 17 19 24 23 29)(4 44 46 40 18 20 22 30 28)
= (12 50 14 52 32 26 34 42 48)(13 51 31 27 33 43 47 41 49)
are automorphisms of G.
Note that the automorphism α fixes the vertex 1 and permutes cyclically its
neighbours 2, 42 and 0. Thus, in order to show that the graph is edge-transitive
it is sufficient to show that any odd-numbered vertex can be mapped into 1 by
an automorphism of G. This can be done by appropriate products of α and β.
For example, α4β maps vertex 53 to vertex 1.
However, the graph is not vertex-transitive because from an odd-numbered
vertex it is possible to have three different paths of length 4 joining the vertex
to some other common vertex (for example, vertex 1 to vertex 5), but this is
not possible from an even-numbered vertex.
Another way to show that the Gray Graph is not vertex-transitive is to con-
sider the distance sequences of its vertices [166]. The distance sequence of a
vertex v is the vector (a0, a1, . . . , ar) where ai is the number of vertices at dis-
tance i from v. In the case of the Gray Graph, the distance sequences of the
vertices in the two colour classes are (1, 3, 6, 12, 12, 12, 8) and (1, 3, 6, 12, 16,
12, 4), respectively, therefore these vertices cannot be in the same orbit under
the automorphism group of the graph.
Although the Sage package has the Gray Graph already implemented, it is
easy and instructive to show how to construct it following the specification
given earlier. First one creates the vertex-set which will be the list of numbers
from 1 to 54. Sage, like many computer languages such as Python, starts its
lists from 0. Therefore a list of length n produced by the command range(55)
would contain the numbers from 0 to 54. To start the list from 1 we have to
define the list of vertices as
vertices := range(1,55);](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-41-320.jpg)
![22 Various Types of Graph Symmetry
The graph is then constructed first using the command DiGraph so that
we do not need to repeat every pair of adjacent vertices twice. This com-
mands basically takes two parameters. The first parameter is the vertex-set
of the graph to be constructed, and the second parameter is a boolean function
(defined with the lambda construct) of two variables which compares all pos-
sible pairs of the vertex-set and an edge is drawn between any pair of vertices
for which the function returns True.
dgray := DiGraph([vertices, lambda i, j:
(i == mod[j + 1, 54]) or
(i == 53 and j == 54) or
(i == 1 and j == 42) or
(i == 2 and j == 15) or
(i == 3 and j == 28) or
(i == 4 and j == 33) or
(i == 5 and j == 44) or
(i == 6 and j == 53) or
(i == 7 and j == 48) or
(i == 8 and j == 21) or
(i == 9 and j == 32) or
(i == 10 and j == 45) or
(i == 11 and j == 24) or
(i == 12 and j == 41) or
(i == 13 and j == 20) or
(i == 14 and j == 31) or
(i == 16 and j == 35) or
(i == 17 and j == 40) or
(i == 18 and j == 49) or
(i == 19 and j == 54 or
(i == 22 and j == 51) or
(i == 23 and j == 30) or
(i == 25 and j == 38) or
(i == 26 and j == 43) or
(i == 27 and j == 52) or
(i == 29 and j == 36) or
(i == 34 and j == 47) or
(i == 37 and j == 50) or
(i == 39 and j == 46) ] )
This digraph is then changed into an undirected graph with the following
command which changes every arc into an edge.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-42-320.jpg)
![2.2 Asymmetric graphs 27
Define a graph Hρ with vertex-set the pairs in Eρ and such that each pair
{v, w} is adjacent to the pair {ρ(v), ρ(w)}. In Hρ, all degrees are at most 2.
Degrees equal to 1 could arise because {ρ(v), ρ(w)} could contain an element
not in W, and so the pair would not be in Eρ. Degrees equal to 2 could arise
because {v, w} could be adjacent to both {ρ(v), ρ(w)} and {ρ−1(v), ρ−1(w)}.
Therefore the components of Hρ are isolated vertices, paths or cycles. Let Iρ
be a set of independent (that is, mutually not adjacent) vertices in Hρ. There-
fore, for any pair {v, w} ∈ Iρ, {ρ(v), ρ(w)} is not in Iρ.
Now, all isolated vertices in Hρ are independent, at least half of the vertices
on a path are independent and at least one third of the vertices on a cycle are
independent, the extreme case here being a triangle. Therefore
|Iρ| ≥ |Eρ|/3 ≥ 2g(t − 2)/6,
as required.
Corollary 2.6 Let G ∈ G(n, 1
2 ), W ⊂ V = V(G) and |W| = t. Let ρ : W → V
be an injective function that is not the identity. Let g = g(ρ) be the number of
elements w ∈ W such that ρ(w) = w. Let Sρ be the event
‘ρ gives an isomorphism from G[W] to G[ρ(W)]’.
Then
P(Sρ) ≤
1
2
2g(t−2)/6
.
Proof Let Iρ be the set constructed in the previous lemma. Now, for a given
pair {v, w} ∈ Iρ, the event
‘{v, w} and {ρ(v), ρ(w)} are both edges or nonedges’
has probability 1/2. These events, as they range over all pairs {v, w} ∈ Iρ, are
mutually independent because they involve distinct pairs. But Sρ requires all
these events simultaneously. Therefore, by independence,
P(Sρ) ≤
1
2
|Iρ|
≤
1
2
2g(t−2)/6
,
as required.
The result of this corollary is the crux of the proof of the following theo-
rem: There are too many independent correct ‘hits’ required for ρ to be an
isomorphism, and the probability therefore becomes small as n increases.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-47-320.jpg)
![28 Various Types of Graph Symmetry
Theorem 2.7 (Korshunov; Müller; Bollobás) Let k be a fixed nonnegative
integer and let G ∈ G(n, 1
2 ). Let pn denote the probability that
∃W ⊆ V(G) = V = {1, 2, . . . , n},
with |W| = n − k and such that
∃ρ : W → V, ρ = id, ρ is an isomorphism from G[W] to G[ρ(W)].
Then, limn→∞ pn = 0.
Hence, a.e. graph has property Ak.
Proof Pick a particular W ⊂ V with |W| = n − k. This can be done in
n
n−k
ways, and
n
n − k
=
n(n − 1) . . . (n − k + 1)
k!
nk
.
Let t = n − k. Let ρ : W → V be injective and not the identity, and let
g = g(ρ) be the number of vertices of W that are moved by ρ. Let Sρ be the
event defined in the previous corollary.
Now, for a given value of g between 1 and t, how many functions ρ are there
such that g(ρ) = g? Such a function is determined by the set {w : ρ(w) = w}
and by the values it takes on this set. Therefore, there are less than n2g such ρ.
Therefore, for a given fixed W, the probability of a nontrivial isomorphism is
given by
ρ=id
P(Sρ) =
t
g=1
ρ:g(ρ)=g
P(Sρ)
≤
t
g=1
n2g
1
2
2g(t−2)/6
=
t
g=1
n2
2(2−t)/3
g
t
g=1
41/3
n2
2−t/3
g
.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-48-320.jpg)
![“Die olie word verkry uit die bessies van hierdie boom deur dit fyn te stamp en
dan te kook; die bessie is net so groot as die kasterolieboom se pitte, dog is
baie harder. Hierdie bessie braai Rondelyf uit saam met die melk van die gifbos.
As dit afgekoel en koud is, dan is dit gereed om as olie vir verf te dien. Waar die
soort gekookte olie op ’n klip val, maak dit ’n bruin vlek wat nie vandag, nòg
môre, nòg oor jare daarna somar sal uitgaan nie.
“My oorlede moeder het ons ou trektafeltjie met daardie soort verf geskilder. Op
sagte voorwerpe, soos sagte hout, word die verf dof, maar op harde klip behou
[145]dit sy helder kleur. Maar met al ons rondtrekkery het dit moeilik gegaan om
’n skaafplekkie aan die tafel te kry—dis ongetwyfeld, dié soort olie is goed.
“Rondelyf het nog ’n ander soort verf gemaak, naamlik van ajoos, vet en
harpuis. Die Boesmanlandse harpuis is baie skaars—so vat hy soms in plaas
daarvan kors, of gom, van die t’kooibos. Met hierdie soort verf het hy soms sy
hele liggaam besmeer—tot selfs sy bakkies. Dit het ’n mooi donkerbruin kleur.
“Die ajoos is ’n soort plant nes die duiwelsbrood. [In Boesmanland word ’n
paddastoel duiwelsbrood genoem.—Die Skrywer.] Maar waar die duiwelsbrood
die vorm van ’n hoed of sambreel het, het die ajoos die vorm van ’n ronde bal.
As dit ryp word, dan droog die vel uit, word hard; en as die vel dan oopgebreek
word, dan is die hele bal van binne vol van ’n bruin, swart of geel poeier.
Hiervan word die verf gemaak.”
Tot so ver het oom Jan vertel.
Die ajoos word in die Karoo en binnelande van die ou Kolonie oeltjie, of
nambossie, genoem; en word sowel deur die Hottentotte as Boesmans gebruik
om hulle wange mee bruin te skilder—soms word hulle gesigte nogal taamlik
bont daarmee geverf.
Ons het in ons kinderdae gesien dat blank meisies ook hulle gesigte tydelik
daarmee geskilder het. Toe ons die rede daarvoor verneem, was die antwoord:
“Dit hou die vel koel en as ons dit na ’n tyd afwas, dan is ons gelaatskleur weer
lelieblank; want hier in die ope veld [146]brand die son ons heeltemal bruin.” Dit
was natuurlik die geval waar die trekboere ver uit mekaar staan en besoekers
nie aldag daar kom nie. Die meisies hou hul by sulke geleenthede eenkant.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-50-320.jpg)
![VOORBEELDE VAN VERFKLEURE.
Wel, om tot die verfkleure van die Boesmans terug te keer. Boesmans teken die
Kaffers altyd swart af—daarvoor gebruik hul potswartsel; vir Boesmans gebruik
hul meestal bruin—daarvoor gebruik hul bruin potklei, bruin klip wat ysterroes
bevat; vir wit mense gebruik hul wit of liggeel klei, of skilferklip van dié kleur.
Ook het ons skilderinge in groen en blou gesien, wat vermoedelik aan kopererts
ontleen is. Soms vind ’n mens tekeninge wat amper rooi is; die verf hiervan
word op seker plekke gegrawe. [149]
[Inhoud]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-51-320.jpg)
![Kinders van die Natuur.
OPMERKINGS:—In hierdie Verhaal deel ons ’n paar gevalle mee hoe onverskillig ’n
Boesmans die erns van die lewe kan opneem: plesier is sy grootste ideaal.
Baie mense eet om te lewe; ander, weer, lewe om te eet en om die plesier van
die wêreld tot hoofdoel te stel. Dit lyk of die Boesman hom beskou as net in die
wêreld te gekom het om te eet en plesier te maak.
Sy grootste genot is dan maar eet en dans. Hy kan nie lekker dans as hy honger
het nie; en as hy sy bekoms aan ’n maal het, dan wil hy dans; en dit doen hy so
goed dat almal wat hom sien dans, moet verwonderd staan. Dit het baiekeer
gebeur dat as die baas van die plaas kuiergaste van ver kry, dat hy sy Boesmans
roep om tot vermaak van die kuiergaste te kom dans—net soos die ou Farao’s
van Egipte gedoen het om hulle dwergdansers tot vermaak van ander en
hulleself te laat dans.
Sê die oubaas: “Toe, ou Hans, kom dans vir die kuiermense die aasvoëldans,”—
dan lyk ’n aasvoël ’n mooi bog om beter rond te spring en met sy vlerke rond te
klap as ou Hans Boesman dit kan doen. Dan kom weer die bobiaandans,
wanneer ou Hans Boesman se litte so los is as dié van ’n lamlendige houtpop.
Daarna moet hy die [150]springbokdans uitvoer, en dan by hierdie geleentheid
wip Hans soos ’n gomlastiekbal in die lug, skynbaar sonder ’n been, arm of spier
te beweeg—want springbokke hou mos hulle bene en nek styf as hulle pronk.
En as ’n klugspeler en nabootser het ’n Boesman nog nie sy gelyke gekry nie.
Hy is ook glad nie punteneurig omtrent die soort kos wat hy kry nie. Dit kan
maar goeie of slegte vleis wees, slange of skilpaaie, veldvrugte of uintjies—alles
is welkom, as dit maar net nie doodmaak nie.
Maar dan gebeur dit met groot droogtes dat kos skaars word en hulle hard moet
stry om elkeen vir homself aan lewe te hou; dan maak hulle wonderlike planne
om kos in die hande te kry. Loop hul dan kos op die lyf wat vir ’n week of meer
kan dien, dan is daar geen voorsorg om matig aan te gaan en ’n stukkie vir die
dag van môre opsy te sit nie. Nee, hulle eet gulsig tot hul nie meer kan nie,
gaan slaap as dit dag is, of dans as dit nag is. Is alles op, dan bly die gedagte
oor: “Ons het dit die dag of wat lekker en plesierig gehad.”](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-53-320.jpg)
![As honger en droogte diere teister, dan gooi hul hulle kleintjies weg; en wat
doen die Boesmans?
Mevr. Jacob de Clercq vertel aan ons dat toe sy nog ’n kind was, kom daar na
haar vader, Gert van Niekerk, se plaas (waar nou Uitkyk-stasie, Middelburg, Tvl.,
is) ’n ou Boesman en sy meid; hulle plaas ’n klein meidjie wat nog nie kan loop
nie, voor haar moeder en vader en soebat om tog maar die meidjie vir ’n jong
os te ruil, daar hulle [151]baie honger ly. Ou mnr. Van Niekerk en sy vrou wou so
’n jong kind nie ontvang nie; dog die Boesmans bly aan soebat, hulle verlaat die
werf stilletjies, gaan ’n bees vat en gaan daarmee voort. Mnr. Van Niekerk het
die kind met ’n oorlamse Kaffer agternagestuur, maar hy kon die Boesmans nie
vind nie—so het mevr. Van Niekerk maar die meidjie grootgemaak. Sy het haar
ounooi getrou gedien en het tot haar oudag by die famielie Van Niekerk
gewoon, en het naderhand sleg van gesig geword.
Maar dit moet al baie droog wees as Boesmans nie meer iets te eet in die veld
kry nie. Laat ons weer hoor wat oom Jan Visser ons vertel. Mnr. De Roubaix
stuur die volgende mededeling aan ons:—
Oom Jan Visser vertel: “Ek was eenmaal as kind met my vader mee in die
Kalaharie-woestyn. Daar het ons eendag—kort voor sononder—’n klomp
Boesmans teen ’n duin sien koes en opstaan. Hulle was druk besig om iets te
soek en uit te grawe en steur hul nie aan ons nie. Rondelyf weet dadelik te
vertel dat die Boesmans honger het en dat hul besig is om uintjies te grawe. Toe
sien ons dat kort-kort een op sy hurke gaan sit en druk besig met grawe is. Dit
het waarlik gelyk nes ’n klomp bobiane wat kos teen die rant soek.
“Toe dit begin skemer word, het die Boesmans teen die duine gaan rus.
Rondelyf vra toe somar verlof om daarheen te gaan, daar hy vermoed dat die
hele nag fees sou gevier word. Hy was reg; want so was dit ook. Die Boesmans
het hulle ramkies meegebring en het daarop die [152]hele nag gespeel. Die
gedans was net lewendig en geesdriftig. Ons kon die musiek duidelik hoor, en
daar is een riel wat hul toe gespeel het, wat ek self op Rondelyf se ramkie kon
speel. Die naam van die riel is Die Bontperd. Dit het net twee of drie draaitjies
en gaan min of meer so: Hor-tiek-tiek, hor-tek-tek, hor-tak!—dit word oor en oor
herhaal.
Toe Rondelyf terugkom, vertel hy dat van hierdie Boesmans het die poliesie van
Upington en Kenhardt vroeër gaan haal en met die grootste gesoebat op](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-54-320.jpg)
![Upington gekry. Daar is hul toe gehuur geword om êrens anderkant Kenhardt
aan ’n dam te gaan werk. Maar voor hulle die werkplek bereik het, het die laaste
een weggeloop en die Kalaharie ingevlug.
“Toe ek ’n kind was, het Rondelyf vir my meer as een ramkie gemaak. Hy het
gewoonlik die hoepel van ’n vat geneem en rond gebuig dat dit so groot as ’n
hoed se bol is. Aan die hoepel bind hy ’n stok wat as nek van die ramkie dien;
bo-aan maak hy drie gaatjies waarin die pennetjies moet kom wat die snare
moet aandraai. Hy neem dan ’n nat haarafgemaakte vel van ’n lam en trek dit
oor die rondgebuigde hoepel en oor die stok so ver dit aan die hoepel vas is.
Dan maak hy die drie snare aan die onderpunt van die stok vas, trek dié oor die
romp van die ramkie en bind dit een vir een aan sy pennetjie vas. Dan maak hy
’n kam en plaas dit op die romp van die ramkie; dan lyk die affêre nes ’n drie-
snarige ghitaar.
Nou is ou Rondelyf my musiekonderwyser—ja, wat vang kinders nie aan nie! Ek
het geleer om Die Bontperd [153]te speel, dan dans ou Rondelyf voor my en
skoffel dat die stof so trek.”
Tot so ver die verhaal van oom Jan.
’n Boesman sal hom die hele dag vreeslik vermoei om in die aand—al was dit
maar vir ’n oomblikkie—’n plesier-plek by te kom. Ons weet van ’n geval waar ’n
smous ’n mak springbok wat met die vee meeloop, geruil het. Die springbok kry
dit in sy kop om terug te hol om sy pleegmoeder, die bokooi, te gaan opsoek. ’n
Boesman bied sy dienste aan om die jong springbokkie vir ’n sopie brandewyn
te gaan haal. Die smous beloof hom dit, en kort na sonop spring die Boesman
weg en kom moeg die aand na sononder met die bokkie op sy skouer terug. Toe
iemand hom vra of hy sy lyf dan oor ’n sopie so moet pla, antwoord hy: “Baas,
’n dag se swaarkry is niks, as ek net in die aand effe my lyf kan plesierig maak.”
Die smous het egter ’n span roltabak by die sopie gevoeg; en meer voldaan en
opgeruimd—toe die sopie begin te trek—het ons ’n skepsel nie gesien nie. Met
so ’n beloning sal hy nooit staak nie.
Oom Piet Smit het ’n Boesman en sy seun gehad wat snags na die skape in die
bosveld moes kyk. Hulle weet as hul durf die veewerf in die nag verlaat, dan kry
hul die volgende dag ’n afgedankste afranseling. Maar daarom gee hul nie as hul
net in die nag die bierpotte van die Kaffers wat op ’n afstand woon, kan bykom.
Hul steel enigiets om vir bier te gaan verruil. [154]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-55-320.jpg)
![Jagters in die Kalaharie kry baie dienste uit die swerwende Boesmans net deur
hul met ’n stukkie tabak te vergoed. Doen hul dit nie, dan moet hul van dors
omkom. Dis wonderlik dat die wildste Boesman van tabak gehoor het en baie uit
sy pad sal gaan om dit in hande te kry. [157]
[Inhoud]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-56-320.jpg)
![Fyn Diefstalle.
OPMERKINGS:—Nou sal ons enige staaltjies gee hoe fyn ’n Boesman kan steel sonder
uitgevind te word—maar baiekeer loop hy hom tog vas, daar die witmense te goed
met sy planne bekend is.
Daar is baie mense wat volhou en nie van die punt af te kry is nie, nl. dat ’n
Boesman ’n gebore dief is. En tog sit in so ’n algemene verklaring nie altyd
waarheid nie. Hoeveel mense vertrou hulle goed aan die sorg van Boesmans toe
en vind hulle vertroue nie misplaas nie? Ons was in ’n huis waar die
Boesmanmeid aan die nooi sê: “Nooi, ek sal my hande nie aan jou goed sit nie;
maar ek beloof nie om nie te steel as ek by jou drank uitkom nie.” En die
huisvrou het ons verseker dat nie ’n speld verlore geraak het nie, maar die drank
kon sy nooit te veilig onder slot gehou het nie.
Andermaal gaan ons vertel wat oom Jan Visser verhaal het en aan ons deur mnr.
E. de Roubaix toegestuur is:—
“Eenmaal het ’n veekoper met sy vee by ons staanplek gelê. Die vee kom
gewoonlik oor ’n kaal vlakte met klein bossies begroei na hulle lêplek toe. As ’n
haas oor die veld loop, kan ’n mens hom sien. Wel, soos gewoonlik kom die
[158]vee weer een agtermiddag oor dieselfde vlakte wei-wei veewerf-toe; en die
veekoper se twee wagters was by.
“Die volgende oggend, na ou gewoonte, word die skape getel, en toe word
bevind dat een aan die getal kort kom.
“My oorlede vader merk toe teen die veekoper op: ‘Dis baie snaaks dat net een
skaap aan die getal ontbreek—een skaap sal mos nie afdwaal nie! Ons het seker
sleg getel, of anders is die skaap gesteel of aan die geilsiekte dood.’
“Hulle tel weer die skape; maar nog word een vermis. Toe besluit hul om met
die honde effe in die veld rond te soek. Hulle was nog geen paar honderd tree
van die tent af nie of hul was op daardie kort blomkool-vlakte. Hulle sien dat die
honde so gretig in ’n droë bos snuffel. Na nader te gestap het, bevind hul dat dit
’n droë kraalbos is wat iemand in ’n erdvarkgat se bek geprop het. Hulle trek die
bos uit, loer in die gat—en daar vind hul die skaap afgeslag, en die vleis,](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-58-320.jpg)
![behalwe die stert en rugstring, was in die vel toegedraai en aldus in die gat
sorgvuldig versteek.
“‘Dis niemand anders se werk hierdie nie as dié van Rondelyf,’ roep my vader
uit. ‘Ek sal die hele ding gou-gou vir jou uitvind,’ sê hy toe hulle huis-toe stap.
“Toe hul daar aankom, pak hul Rondelyf aan sy arm en bind sy hande agter sy
rug vas en lei hom na die klein handskroefie om sy oor daarin vas te draai. Want
’n Boesman moet ’n mens gou op die lyf loop om hom bang te maak en nie kans
gee om te dink nie.
“‘Kom, Rondelyf, vertel die hele waarheid; hoe het jy die skaap daar op die kaal
vlakte gevang sonder dat ’n [159]mens of die twee wagters jou gesien het?’ vra
my vader toe hy maak asof hy die skroef met geweld gaan aandraai. ‘Toe, kom
uit met die waarheid, of ek draai jou ore en vingers in die skroef af!’
“Toe kom hy sonder te versuim met die hele waarheid uit, en vertel dat hy in die
agtermiddag al in die erdvarkgat met ’n bossie op sy kop gebind, gaan lê het;
want hy weet dis die huistoekom-pad van die skape. Toe die skape om die
erdvarkgat wei, het hul hom nie daar binne-in gewaar nie. Hy het sy vet hamel
uitgesoek, wat geen tree van die gat af was nie, het sy hand uitgesteek, hom
gegryp en in die gat gesleep. Die skape om die gat het wel ’n bietjie geskrik;
maar dit doen hul heeldag as hul op ’n hasie, slang of skilpad afkom. Dadelik
was alles oor; die skape trek die gat verby sonder dat die twee wagters geweet
het dat daar iemand en ’n skaap binne in die gat gewees het.
“Na hierdie verklaring was Rondelyf gou uit die skroef verlos; dog my vader het
hom ’n afgedankste loesing afgetel omdat hy durf die goed van ’n ander steel.
Dit het vir Rondelyf en sy maats skaars ’n week gehelp, dan het hul die taaiste
pak slae vergeet en steel weer van vooraf.”
Mnr. Piet Venter het ons vertel dat sy vader, toe hy nog in die binnelande van die
ou Kolonie gewoon het, twee getroue honde gehad het. Hulle het baie jakkalse
en ongediertes doodgebyt; en ’n Boesman het nie gedurf om naby te kom nie.
Hulle het ook ’n getroue ou Boesman, ou Ertman, gehad, wat na die koeie met
jong kalwers kyk. [160]
Eendag slag my vader, mnr. Piet Venter, ’n vet koei; die vleis moet in die nag in
’n boom hang, en die twee honde pas dit op. Dog die ander dag makeer ’n](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-59-320.jpg)
![voorkwart; en hul kon nie uitvind hoe dit onder die oë van die honde uit gesteel
is nie. Later, op ’n wonderlike manier, vind hul uit.
Ou Ertman het aan sy maats in die veld vertel dat sy baas ’n koei gaan slag. In
die nag het een in ’n boom ver van die huis geklim en nes ’n jakkals geskree;
solank die een trag die honde na hom, die sogenaamde jakkals, te lok, het die
ander êrens naby verskuil gelê. Die honde hol na die sogenaamde jakkals toe en
blaf daar; onderwyl steel die ander Boesmans die vleis en hol daarmee veld-in
en vlug ver die berge in; dog hulle het nie vergeet om stilletjies in die weiveld
aan Ertman sy beskeie deel van die vet vleis te bring nie. [163]
[Inhoud]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-60-320.jpg)
![As nood die Boesmans druk.
OPMERKINGS:—Hier gee ons ’n paar voorbeelde hoe ver nood iemand kan drywe.
Die boesman, as kind van die woestyn, het ’n harde stryd teen wilde diere,
vyande, honger en dors. Selfverdediging is die eerste wet van bestaan. En daar
is so baie maniere van selfverdediging.
Ons het al vertel hoe Boesmans soms hulle kinders vir kos verruil. Dit is nie
alleen om hulle lewe te red nie, maar ook dié van hulle kinders.
Maar hoe maak hul met hulle swak ou volk wat nie meer met jag kan meedoen
nie as die nood hul dryf om te trek—ja, na ’n veraf geleë streek te trek?
Hieromtrent het ons informasie ingewin van mnr. W. A. van Zyl, Klein-Breipaal,
Boesmanland. Dit deel hy ons mee: “Die wilde Boesmans het drie maniere om
van hulle ou volk ontslae te raak as die nood hul erg druk, sodat hul verplig
word om die streek te verlaat.
“Die eerste manier is om ’n kliphok om die swakke en oue van dae te pak.
Daarin plaas hul dan kos en water om vir ’n geruime tyd te dien. Dan trek hul
weg en laat die ou persoon verder aan sy eie lot oor. Die kliphok, wat hoog en
sterk gepak is, is natuurlik bedoel om die ou skepsel [164]teen verskeur van wilde
gedierte te beskerm. As hy binne-in veilig is, dan word hy van die vrees en
angste gespaar om deur wilde diere wreed gedood en opgevreet te word.
“Die twede manier is: Hulle grawe ’n diep ronde gat regaf—natuurlik waar geen
klippe te vinde is nie—, plaas die ou daarin, en ook kos en water, maar dan vir ’n
kort tyd. As hul dan nie weer terugkom voor hy dood is nie, dan smeer die
meide hulle hare vol verf met vet gemeng. Hulle neem dan klein klippies en gooi
dié rondom sy lyf in die gat totdat hy hom nie meer kan roer nie en daar vassit;
dan verlaat hul die plek, om nie na die graf terug te kom nie. Die ou kan hom
dus nie roer nie en moet so van ellende sterwe.
“Die derde manier is: As hul op trek is en een van die oues beswyk van
ouderdom of dors, dan maak hul ’n langwerpige hol plek in die grond, lê gras
daarin en plaas die ou daarop: daar moet hy lê tot hy dood is—nou is daar geen
kos of water om aan hom agter te laat nie.”](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-62-320.jpg)
![Tot so ver het mnr. Van Zyl ons vertel.
In ieder geval waar hul hulle oues agterlaat, is ons deur Boesmans verseker
geword dat hul altyd ’n kliphok, waar klippe te vinde is, en ’n doring-heining,
waar doring-bosse te vinde is, pak, om die agtergeblewene teen verskeuring
deur roofdiere te beskerm.
Dit alles lyk wreed; maar ’n Boesman het ons die vraag self gestel: “Sal baas die
hart hê om jou eie pa of ma [165]koelbloedig te vermoor as daar net liefde en nie
rusie is nie? Wat help dit om agter by hulle te bly? Dan sterf ons met vrou en
kinders almal tesame—en wat word dan van die Boesman-nasie?”
Omtrent wat rein en onrein is, bestaan by volkere groot verskil. Ons eet b.v.
skaap- of beesboutvleis, snoek, hase, vark, ens., terwyl ’n Jood dit as onrein ag.
Ons eet mossels, klipkous, maar nie seekat nie, terwyl die suidelike nasies van
Europa seekat eet omdat dit aan die skulpsoort behoort. Ons eet ystervark, wat
’n knaagdier is, maar verfoei muise, wat ook knaagdiere is. Ons eet nie perde en
haaie nie, terwyl dit in Europa wel geëet word. Ons eet nie kruipende diere nie,
maar wel skilpaaie. Ons eet nie paddas nie, en in Frankryk is dit ’n lekkerny. En
so kan ons nog baie goed opnoem, b.v. dat Kaffers vis as onrein beskou, maar
hulle sal ’n jakkals eet. Baie mense wil nie wild eet nie of dit moet ’n luggie hê.
Nou, die Boesman steur hom aan niemand nie—hy eet wat voorkom, soos vleis,
vis, muise, paddas, slange, akkedisse, skilpaaie, wurms en vleis, of dit vars of
oud is, miere en alles—as dit net nie doodmaak nie. Hy gaan dus met almal
saam. Want die honger en dors van die woestyn het hom daartoe gedwing—net
soos ons in die geskiedenis lees dat as ’n bemuurde stad beleg is om die
bevolking uit te honger, dan het die mense net alles geëet en gedrink wat ’n
Boesman eet en drink. Maar as hy goeie kos kry, bepaal hy hom meer daarby as
tot die ander, [166]en beskou dan b.v. slakke en paddas in dieselfde lig as die
Franse.
Deurdat ’n stomp pyl—soos dié van ’n Boesman—nie deur die dik vel van groot
wil kan boor nie, is die Boesman verplig om met gifpyle te skiet. Maar hoeveel
lewens van hulle het dit gekos voor hul die kuns goed verstaan het om te weet
watter gif te gebruik en hoe om dit te gebruik? Die nood het hom dus tot gif
gedrywe. [169]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-63-320.jpg)
![[Inhoud]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-64-320.jpg)
![Wat ’n Boesman alles kan doen.
OPMERKINGS:—Ons noem hier enige feite wat ’n ander nasie moeilik ’n Boesman sal
nadoen.
Vra aan iemand wat so te sê aldag met Boesmans omgaan, of hy ooit ’n
Boesman ontmoet het wat verdwaal het; en hy sal ronduit verklaar dat hy van
so ’n geval nie weet nie. As ’n Boesman eenmaal op ’n plek gewees het en weet
waar dié plek geleë is, dan sal hy van enig ander oord reguit met ’n onbekende
pad na die plek toe loop—al was dit in die donker. In hierdie opsig is hy net soos
’n posduif wat oor honderde myle deur mistige weer huis-toe vlie.
So vertel mnr. H. Visagie ons dat toe hy ’n jongeling was, het by sy vader ’n ou
Boesman en sy gesin gewoon. Die ou het maar lus gehad om noordwaarts na
die Kalaharie te trek. Na hy vir maande en maande weg was, kom op ’n goeie
dag een van die ou se kinders aangestap—hy het na sy oubaas toe weggeloop.
Volgens die twaalf-jarige Boesmantjie beduie, het hy met sy ouers eers noord-
wes getrek, toe weer reg sonop; en nadat hy maande met hulle rondgeswerf
het, het hy weggeloop en reguit deur die veld koers na sy oubaas gevat. Hy was
vir dae op pad, het baie honger en dors gely, en is deur ’n streek gegaan waar
[170]leeus en wolwe ronddool. Toe hom gevra was hoe hy die pad reggekry het,
was sy enigste antwoord: “Ek weet mos waar oubaas woon.” Wel, sou ’n kind
van twaalf jaar van ’n ander nasie dit reggekry het om die so onbekende weg te
vind, om honger en dors so te trotseer en om die vermoeienis deur te staan,
sonder eens van die gevaar van ongediertes te praat?
Ons het self gesien hoe hul die warmste van die dag uitkies om wild te agtervolg
—wat beteken dat hul ’n harder natuur besit as ’n dier. So skryf mnr. E. de
Roubaix ons verder wat hy uit die mond van oom Jan Visser aangeteken het:—
“My vader het vir Rondelyf meestal gebruik om verlore vee op te spoor—en
hiermee het hy sy weerga nie kon vind nie. Hy kon spoorsny beter as ’n gewone
hond en kon beter sien as ’n aasvoël uit die lug; want die geringste
verskuiwinkie ontsnap sy blink ogies nie. Daarby het hy ’n groot stuk wêreld in
’n omsientjies kon afdraf sonder enige tekens van vermoeienis te toon. Meer as
een middag, as dit regtig warm is, het hy wilde volstruise van agteraf
ingehardloop en gevang. By hom het ek as kind dikwels volstruisbiltong geëet.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-66-320.jpg)
![“Hy was ’n opregte Boesman; maagseer, koppyn, jig, of tandpyn het hy in sy
hele lewe nie gekry nie. En soos hy is, so is al die Boesmans.”
Ofskoon die Boesman se gesig goed is, is sy ruik enigsins in seker opsig
gebrekkig; daarom kan hy die stank van aas of sleggeworde vleis goed verdra.
Selfs die [171]spelonk waarin hy woon, het ’n walglike walm wat enig witmens
sou terugdrywe. En wat nog meer is, is dat hy in so ’n omgewing nie die pes kry
nie; sou witmense onder sulke omstandighede die lewe moet voortsit, dan lê
almal binne ’n week of so plat aan die koors.
Wat hy nog meer kan doen, is om kaal op nat grond te slaap. Soms is sy
skuilplekkie in ’n renerige nag of dag net om te troos, maar nie om die
instromende water uit te hou nie—en tog weet hy van geen jig of inflammasie
nie.
Ons het reeds vermeld watter fyn planne hy met jag kan maak. Mnr. W. A. van
Zyl, van Klein-Breipaal, deel aan ons mee:—
“Boesmans bekruip wild seer fyn. Solank die wild nog op ’n afstand is, kruip hy
hande-vier-voet; dog kom hy nader, dan seil hy op sy buik. Al wat hy moet
doen, is om ’n bossie op sy kop vas te maak om sy swart hare te bedek; want
dit is al wat die wild kan beken, daar sy lyf dieselfde kleur as die grond het. Hy
is dan in staat om so na aan die bok te kom as hy verkies—net, hy moet
onderkant die windkoers bly, sodat die bok die bekruiper se ruik nie kry nie.
Dieselfde plan volg hy as hy wild voorlê. Hy kruip dan gewoonlik tot op ’n
afstand tussen vyftien en vyf-en-twintig tree, hy kan deur sy fynheid nog nader
kom, maar die pyl moet ’n seker distansie trek om sy snelheid te kry.”
Iets wat ’n Boesman ook beter as ’n ander kan doen, is om weg te kruip. Eens
het ons ’n plaas gemeet waar twee Boesmantjies kalwerwagters was. In ’n
grasvlei het ons ’n vlag gesteek en het die Boesmantjies belet om met die
[172]kalwers naby die vlag te wei. Hulle het hul egter daaraan min gesteur, en
die vlag is verskeie male deur die kalwers uit die grond gepluk geword. Die baas
se twee seuns en ons persoonlik sien die Boesmans in die grasvlei rondspeel;
ons al drie stap daarheen—maar daardie Boesmantjies kry, was net so goed om
kwarteltjies daar te soek, en tog, die gras was nie hoër as twee voet nie. Ons
het vir die pret lank gesoek, maar moes so onverrigtersake omdraai. Toe ons
omtrent aghonderd tree weg was, sien ons hul daar weer rondspeel.](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-67-320.jpg)
![Hierdie eienskap help die Boesman om as jagter uit te blink, en om in ’n
minderheid teen ’n oormag te veg. Hierdeur was hy in staat om sy grond vir so
’n lang tyd teen Hottentot, Kaffer en witmense te verdedig. Eenmaal het ’n ou
Kaffer op die westelike grense van Transvaal hom aldus op sy eienaardige
manier aan ons uitgedruk:—
“My baas, die Boesman is somar die ding. Jy sien, baas, jy en ek maak die
baklei: jy kyke my oog, ek kyke jou oog. Maar die Boesman!—nee, my baas. Jy
pla net verniet jou oog om vir hom te soek. Soos jy soeke die wind, jy kry vir
hom; soos jy soeke die bobbejaan, jy kry vir hom; maar die Boesman! hoe jy sal
weet hy is daar? Ja, al wat jy sal wete is: hy is daar, en hy stuur die dood na jou
toe.”
Die Boesman het die jagveld en die waters as sy eiendom beskou. Die wild is sy
vee en dit word hom afgeneem; so moor hy ook onder die vee van ander. Hy
weet vir hom is daar geen genade nie; waarom moet hy [173]die lewe van ander
in ’n geveg spaar? Hy of sy vyand moet dood. As daar maar ’n dosyn vegmanne
in ’n bergstreek woon, dan is dit nie elkeen se werk om hulle daaruit te
verdrywe nie; en wie sal hom in die woestyn gaan agtervolg?
Sy geheue is baie goed om plekke te onthou waar hy die doppe van
volstruiseiers vol water begrawe het. Dit bly daar lank verberg; want mnr. W. A.
van Zyl skryf aan ons:—
“Hulle woon gewoonlik ver van water. Die water word in volstruiseierdoppe
aangedra na hulle werf. Ook begrawe hul orals op sagte plekke sulke doppe vol
water om in tyd as hul vlug of trek te gebruik. Ek het self in my kinderdae op
sulke gevulde doppe afgekom as die wind hul in die duine oopwaai.”
Ja, en ons wonder hoe water wat solank in ’n eierdop gestaan het, moet smaak!
Ons sal in die volgende boekdeel ander ware gebeurtenisse meedeel wat haas
ongelooflik is, en sal aantoon wat ’n Boesman nog meer kan doen. [177]
[Inhoud]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-68-320.jpg)
![Wat ’n Boesman nie kan doen nie.
OPMERKINGS:—Uit hierdie Verhaal blyk dit dat die Boesman steeds ’n raadsel vir die
ontwikkelde mens bly; hoe meer ons hom bestudeer, hoe raadselagtiger is hy.
As nasie van jagters wat net leef van wat hul gedurende die dag versamel het
en wat hul meen die volgende dag weer in te samel, en ook dat hul geen vee
van hulle eie aanhou nie, so het die Boesman hom nie erg bekommer oor getalle
nie. En daar hy van getalle min verstand het, kan hy ook nie syfer of reken nie.
En tog, daar sal geen pyl van hom wegraak nie, of hy weet dit. As hy bokwagter
is, sal daar geen bok vermis raak nie, of hy weet dit. Al lyk lammertjies hoe
eenders, tog weet hy om aan iedere ooi haar regte lam te besorg. Van die
geldwaarde van iets het hy hoegenaamd geen begrip nie; en as ’n mens hom
geldstukke wys en die name en waarde daarvan opnoem, sal hy dit wel leer;
maar meng ’n spul silwergeld, soos trippense, sikspense, sielings,
tweesielingstukke en halfkrone deurmekaar en vra hom hoeveel dit alles tesame
uitmaak, dan is hy nie in staat om die waarde daarvan op te gee nie.
Daarom word Boesmans vir kos, tabak en ou klere gehuur—hulle verkies dit bo
geld, waarvan hul niks verstaan nie. [178]
Ek het eenmaal ’n Boesman gehuur wat ’n hele ruk op Kimberley gewerk het. Hy
kon daar met die wette maar nie regkom nie, en was daar tweemaal in die
tronk; so is hy daar weg. Ek sê aan hom dat ek hom kos en tabak sal gee en vyf
halfkrone in die maand—ek wys hom op my vingers hoeveel vyf is. Hy skud sy
kop en sê: “Nee, baas, ek het nog altyd op Kim’ley meer verdien; die base daar
het elke maand by elke vinger ’n sieling neergesit; en as baas dit ook wil gee,
sal ek by baas mooi werk; so nie, dan gaan ek ’n ander baas soek.” So het ek
hom vir tien sielings in plaas van twaalf sielings en sikspens per maand gehuur.
En hy het my baie trou gedien; maar dat ek aan sy verstand kon bring dat vyf
halfkrone meer as tien sielings is, was net verniet.
Wie die Goewerment is, kon hy maar nie verstaan nie. Hoe ek hom ook al die
vorm van goewerment uitlê, hy kon daar geen begrip van kry nie—hy meen dit
moet ’n ryk man wees wat alles besit en oor algar baas is. Hy vertel toe:—
“Met ons Boesmans daar in die veld is dit só gesteld: Niemand is oor ons baas
nie. Net ons pa en ma as ons nog kinders is; maar hul sal vergeet om aan ons](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-70-320.jpg)
![te vat as ons self ’n mannetjie is wat met ’n kierie ’n breekskoot kan slaan. As
iemand oor ons wil kaptein speel, nes by die Hottentots en Kaffers, pak ons vir
ons eenvoudig in en laat hom waar hy is. Hy sal dit vir eenkeer en tweekeer
probeer, maar vir die derde keer sal hy gou leer dat hy nie met kinders te doen
het nie. Is dit dan nie reg nie, baas? Kyk, al die wild is mos ons algar syne
saam, die veldkos [179]is ons syne saam, die water is ons syne saam; en wie het
hom daaroor baas gemaak? Al wat ons ons eie noem, is ons vrou en kinders,
ons hond en gereedskap—daar moet ’n ander die pote van afhou: dis ons
eiendom—nie waar nie baas? Dis mos reg so, nè baas?”
Hieruit is dit duidelik dat ’n Boesman nie kan regeer nie en is self moeilik om te
regeer—mits ’n mens sy maniere goed bestudeer en hom daarna behandel; dan
is hy getrou aan sy baas en nooi.
Om ’n Boesman te leer lees en skrywe, is net tydverkwisting. Toe ons persoonlik
veewagtertjie was, het ons smiddags onder die bome in die somer baie tyd
gehad om na Hans Boesman se stories en vertellings te luister. Ons het letters in
die sand gemaak om hom ’n begrip te gee hoe die witmense lees en skrywe. Hy
wou ook probeer om die letters te leer, en het nogal behae daarin geskep; maar
hy moes dit opgee: “dis tog te swaar om te leer,” het hy gesê, en so was die
skoolganery op ’n end; terwyl ons Basoetoe-skaapwagter binne ’n paar maande
uit sy boek kon lees. Dit het ons van persoonlike ondervinding; want ons kinders
het vir Hans as ’n interesante speelmaat beskou en wou hom aanhelp.
In die jaar 1799 het eerw. J. J. Kicherer van die Londense Sendinggenootskap
aan Sakrivier, Boesmanland, ’n sendingstasie aangelê, met die spesiale doel om
die geestelike belange van die Boesmans te bevorder. Aan hulp het dit nie
ontbreek nie. Die boere het beeste, skape en tabak aan die sendeling gegee om
daarmee die Boesmans na Blyde [180]Vooruitzigt—soos die naam van die
sendingstasie was—te lok. Die Boesmans het by hope gekom en daar gebly tot
daar nie meer beeste, skape en tabak gekom het nie; toe het hul weer na die
wilde lewe die toevlug geneem; want werk—soos Hottentots en Kaffers by
sendingstasies—wou hul nie.
Nou, dis gewoonlik dié Boesmans wat ’n tyd onder die witmense gewoon het en
wat weer die wilde natuur aanneem, wat die groot kwaaddoeners is. Mnr. W. A.
van Zyl, Klein-Breipaal, noem hul wildehonde. Dit skryf hy aan ons omtrent die
verwoesting deur hulle aangerig:—](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-71-320.jpg)
![“Die Boesman se geaardheid is nes dié van ’n wildehond. As hulle vee gesteel
het, dryf hul dit op ’n draf vinnig weg tot by ’n ruie plek of klipkoppe, en dan
maak hul almal, grootvee sowel as klein vee, voor die voet op ’n hoop dood. So
het hul troppe en troppe vee van ons doodgemaak—so maak wildehonde ook.
Die wilde Boesman sorg nooit vir die dag van môre nie: wat hy nie kan opeet
nie, kan maar bederwe en vergaan; môre doen hy weer dieselfde.”
Dit het wel gebeur as vee te moeg word en nie vinnig genoeg na die sin van die
Boesmans voortgaan nie, dat hul dan sulke diere die haksenings afsny of hul
doodmaak; maar gesond wil hul die dier nie laat agterbly nie. Dit het ook
gebeur dat hul van die dooie vee vleisskanse maak om agter te lê en hulle so te
verdedig. Wanneer daar so ’n oorvloed van kos is, dan gun hul die leeu ook
daarvan, wat ook vleis vir hulle uit sy oordaad laat bly—die wolwe [181]en
jakkalse kan ook maar snags kom bysit. Wat dan nog oorskiet, is die volgende
dag die Boesman syne—hy is nie inhalig nie, en gun aan die ongedierte ook iets.
Wel, as ons al sulke dinge in oëskou neem, dan laat dit ons ver dink; want ons
sien dat ons hier met die voorgeskiedenis-mensdom te doen het. Dit het
duisende jare geneem om die barbare uit die mensdom te beskawe. En hoe sal
ons dit regkry om meteens van ’n lae, gesonke mensheid in een geslag ’n
beskaafde mens te maak? Beskawing klim van een geslag op na die volgende.
En tog, met dit al is die Boesman op seker punte van vooruitgang vir die Kaffers
en Hottentots voor, soos bv. hulle begrawe altyd hulle dooies (met gesig sonop),
terwyl Hottentots hulle dooies in erdvarkgate stop, en sommige Kaffers die lyke
êrens plaas sonder om dié te bedek. In tekenkuns het hul ook verder gevorder.
[185]
[Inhoud]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-72-320.jpg)
![Wat alles in voordeel van Boesmans is.
OPMERKINGS:—Ons gee nou seker punte aan wat steeds gedien het om die bestaan
van Boesmans vir soveel eeue—nee, duisende jare—te waarborg.
Daar bestaan geen nasie op aarde nie, of hulle voorouers was vir duisende jare
terug barbare wat sonder klere rondgeloop het, wat geen ander werktuie kon
maak nie as van klip, hout en horings; hoe om metale te bewerk, het hul niks
van afgeweet nie. Sommige nasies het eerder as andere tot beskawing gekom,
soos die ou Egiptenare, Babiloniërs, Feniesiërs, Grieke en Sinese; dog die
Engelse, Franse, Duitsers, Hollanders, Russe en Skandinawiërs het later gevolg,
en het toe voordeel van die Griekse en Romeinse beskawing getrek.
Wat ook al mag gegis word deur geleerdes omtrent die oorspronklike
geboorteland van die Boesman, dit weet ons as ’n onbetwiste feit dat Afrika
hulle land was en is. En Afrika, met sy groot inlandse woestyn, het die
noordelike beskaafde volke teruggehou; so het die Boesman die voordeel gehad
dat hy sy rol eenkant kan vry uitspeel.
’n Ander voordeel vir hom is dat Afrika die rykste wêrelddeel vir wild is. Hy het
dus nie nodig gehad om met mak diere opgeskeep te wees nie. Ook die veld het
genoeg wilde vrugte en veldkos vir ’n jagter opgelewer, [186]sodat hy nie nodig
gehad het om self te plant en te saai nie. Het hy lus vir vis, dan maak hy
eenvoudig fuike van latte en lê dié in die riviere, en hy kry oorgenoeg.
Op hierdie manier het hy verkies om in ’n lui-lekker-land te woon. Daar is niks
verédelends in jag nie—dit beteken die vermoor van onskuldige lewens; en om
’n naam te maak, word soms meer wild gedood as wat vir die behoefte nodig is.
So het die Boesman hom nie uit sy barbaarsheid verhef nie en het stil bly staan
sonder verstandelik te ontwikkel; want ’n redelose dier, soos ’n leeu, kan ook jag
en genoeg vind om van te lewe.
Dan is dit ’n natuurwet wat die Skepper daar gestel het, dat die geskikste altyd
sy pad en bestaan beter vind as ’n minder geskikte. Die geskikste ontwikkel
kragtiger en laat meer nakomelinge na as die minder geskikte, wat nie behoorlik
vir sy nakomelinge kan sorg nie. Dit op die Boesman toegepas, is dit duidelik dat
die vader wat die beste kan hardloop, beste kan sien en die gepaste kleur van
vel het om hom net soos die veld te laat lyk—dié vader bring ’n kragtiger](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-74-320.jpg)
![famielie op; terwyl die swakker en siekliker man miskien nie eens ’n vrou kan
kry nie; en kry hy een, dan is sy swakker kinders nie bestand teen die harde
eise van die woestyn nie: hulle sterf en laat ook nie ’n swak geslag na nie. So
het net bly voortlewe wat vir die lewensstryd geskik is.
Laat ons die Boesman nou beskou soos hy vandag is, en let op wat tot sy
voordeel is. [187]
Sy kleur is net soos die veld waarop hy jag. Mense het reeds daarop gelet dat
daar ’n klein verskil in kleur is tussen die Boesmans wat in die Kalaharie woon
en die Boesmans wat suid van Grootrivier gebore en opgegroei is. Daarom het
mnr. W. A. van Zyl in die vorige verhaal gesê dat al wat ’n Boesman nodig het
om te vermom, is sy swart hare, wat deur die wild kan beken word; sy kleur is
net soos dié van die grond.
Aldag is jagdag, maar nie vangdag nie. So gee dit eienskap dat daar dae van
oorvloed is, en dae van gebrek, wanneer honger en dors die jagter en sy
famielie kwel. Hieraan het die Boesman se maag gewend geraak en sy liggaam
het in dié rigting ontwikkel om in tyd van oorvloed voedingskrag op ’n seker deel
op te gaar. Wie ’n Boesman ken, weet dat ’n seker deel van sy lyf besonder fris
ontwikkel is. Dus moet ons die tekenaar nie blameer nie as hy ’n Boesman net
so afteken as hy in die natuur is.
Mense wat in stede en dorpe van kindsdae af woon, kan gewoonlik nie so ver
sien nie as mense wat op die platteland woon, omdat hul meestal in strate loop
wat met geboue begrens is; dus het hul geen ver uitsig nie, en die oog word
geoefen om naby te kyk. Die Boesman, as jagter, moet ver en naby oor die
vlaktes en teen rantjies sy oë laat dwaal; dus is hulle oë van geslagte tot
geslagte geoefen om ver en naby te kyk. Dis daarom dat sy gesig so goed as ’n
vèrkyker is. Hy het dus die voordeel van ’n vèrsiende oog. Sy ooglede vorm
twee skrefies, omdat hy oor vlaktes in skerp lig kyk. [188]
Iemand groot en swaar van gewig kan nie so vinnig hardloop en is ook nie so
rats nie. Slaan maar ag op wat ’n kind in ’n dag afmaak, probeer dit alles na te
doen, en voel of jy in die aand nie doodmoeg gaan slaap nie. Vergelyk jou krag
met dié van ’n kind, en staan dan verbaas. Net so met die Boesman: deur hy
klein is, weeg hy nie swaar nie; die spiere van sy bene is goed ontwikkel. Die
beste hardlopers onder hulle is ook die beste jagters—dus: die beste versorgers](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-75-320.jpg)
![van hulle famielies; en so is hierdie eienskap van geslag tot geslag onder hulle
voortgeplant, net soos dit die geval is met windhonde en resiesperde.
Die hardste en vertroubaarste werkers en arbeiders van die wêreld is persone
met ’n vaste liggaamsbou en ’n gemiddelde grootte van ongeveer ses-voet-
hoogte. Op hierdie eienskap kan die Boesman nie aanspraak maak nie. Dit is
dus wreed om van hom swaar liggaamsarbeid te eis. Maar gee hom vee om mee
te werk, of werk van ’n minder vermoeiende aard, dan is hy daar tuis net soos
enige ander.
Wat die Boesman deur al die eeue van uitroei of uitsterwe bewaar het, is dat
daar selde of nooit ’n getroude meid gevind word wat nie ’n aantal kinders het
nie. Ons oumense het reeds in vroeër dae opgemerk dat die Boesmans so snel
vermeerder. Maar nou in ons dae is hul aan die verminder. Die rede hiervoor is
dat die Hottentots en Kaffers, wat van die noorde die land oorstroom het, die
mans vermoor en die meide vir hulle tot vroue geneem het. En in ons dae wil ’n
Boesmanmeid liewers ’n Hottentot of [189]Kaffer tot man hê. Dit is dus die meide
wat geen nasionale gevoel vir hulle eie nasie besit nie; sy is die groot oorsaak
van dit alles; haar verval is die ondergang van haar volk. Haar volk is in die
minderheid en word deur die ander verag, waaroor sy ’n seker mate van
skaamte gevoel; en dit maak dit vir haar so gemaklik om haar eie nasie te
verstoot en haar met ’n vreemde nasie te vereenselwig.
Maar sou die Boesmanmeid soveel respek hê om trou by haar volk te staan,
saam met hulle die lief en leed te deel. dan sal sy die redmiddel wees om die
bestaan van die Boesman voort te plant. Maar noudat die Boesmanjong nie trou
nie, sal dit nie meer baie geslagte duur nie of die Boesman sal tot die verlede
behoort. [190]](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-76-320.jpg)
![Hierdie Project Gutenberg-e-boek bevat eksterne verwysings. Hierdie
skakels werk moontlik nie vir u nie.
Regstellings
Die volgende regstellings is aan die teks gemaak:
Bladsy Bron Regstelling Redigeerafstand
11 afbeel afbeeld 1
84 skerppuntbeenbyle skerppuntbeenpyle 1
103 Geskeidenis Geskiedenis 2
113 sy my 1
129 [Nie in die bron
nie] ” 1
179 Beosmanland Boesmanland 2
190 [Nie in die bron
nie] . 1
Afkortings
Oorsig van afkortings gebruik.
Afkorting Uitbreiding
b.v. byvoorbeeld
v.K. voor Kristus](https://image.slidesharecdn.com/3906-250505063248-6c65e794/85/Topics-in-Graph-Automorphisms-and-Reconstruction-2nd-Edition-Josef-Lauri-79-320.jpg)
Topics in Graph Automorphisms and Reconstruction 2nd Edition Josef Lauri
- 1. Topics in Graph Automorphisms and Reconstruction 2nd Edition Josef Lauri download https://ebookgate.com/product/topics-in-graph-automorphisms-and- reconstruction-2nd-edition-josef-lauri/ Get Instant Ebook Downloads – Browse at https://ebookgate.com
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- 6. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 312 Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al. (eds) 313 Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) 314 Spectral generalizations of line graphs, D. CVETKOVIĆ, P. ROWLINSON & S. SIMIĆ 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁŘ (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L. M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier–Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A. H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER 361 Words: notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: high dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: the first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds)
- 7. 373 Smoothness, regularity and complete intersection, J. MAJADAS & A.G. RODICIO 374 Geometric analysis of hyperbolic differential equations: an introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N.H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T. C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍČEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al. (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al. (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al. (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) 395 How groups grow, A. MANN 396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA 397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) 398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI 399 Circuit double cover of graphs, C.-Q. ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J. L. RODRIGO & W. SADOWSKI (eds) 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S. V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢĂ & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for graph eigenvalues, Z. STANIĆ 424 Surveys in combinatorics 2015, A. CZUMAJ et al. (eds) 425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) 426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) 427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) 428 Geometry in a Fréchet context, C.T.J. DODSON, G. GALANIS & E. VASSILIOU 429 Sheaves and functions modulo p, L. TAELMAN 430 Recent progress in the theory of the Euler and NavierStokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) 431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL 432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO 433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA 434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA 435 Graded rings and graded Grothendieck groups, R. HAZRAT
- 8. London Mathematical Society Lecture Note Series: 432 Topics in Graph Automorphisms and Reconstruction Second Edition JOSEF LAURI University of Malta RAFFAELE SCAPELLATO Politecnico di Milano
- 9. University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781316610442 c Josef Lauri and Raffaele Scapellato 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. Library of Congress Cataloguing-in-Publication Data Names: Lauri, Josef, 1955– | Scapellato, Raffaele, 1955– Title: Topics in graph automorphisms and reconstruction / Josef Lauri and Raffaele Scapellato. Description: 2nd edition. | Cambridge : Cambridge University Press, 2016. | Series: London Mathematical Society lecture note series; 432 | Includes bibliographical references and index. Identifiers: LCCN 2016014849 | ISBN 9781316610442 (pbk. : alk. paper) Subjects: LCSH: Graph theory. | Automorphisms. | Reconstruction (Graph theory) Classification: LCC QA166.L39 2016 | DDC 511/.5–dc23 LC record available at https://lccn.loc.gov/2016014849 ISBN 978-1-316-61044-2 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 10. Lil Mary Anne, Christina, Beppe u Sandrina A Fiorella
- 11. London Mathematical Society Lecture Note Series: 432 Topics in Graph Automorphisms and Reconstruction JOSEF LAURI University of Malta RAFFAELE SCAPELLATO Politecnico di Milano
- 13. Contents Preface to the Second Edition page xi Preface to the First Edition xiii 1 Graphs and Groups: Preliminaries 1 1.1 Graphs and digraphs 1 1.2 Groups 3 1.3 Graphs and groups 7 1.4 Edge-automorphisms and line-graphs 10 1.5 A word on issues of computational complexity 13 1.6 Exercises 15 1.7 Notes and guide to references 17 2 Various Types of Graph Symmetry 18 2.1 Transitivity 18 2.2 Asymmetric graphs 25 2.3 Graph symmetries and the spectrum 29 2.4 Simple eigenvalues 31 2.5 Higher symmetry conditions 32 2.6 Exercises 35 2.7 Notes and guide to references 38 3 Cayley Graphs 39 3.1 Cayley colour graphs 39 3.2 Frucht’s and Bouwer’s Theorems 42 3.3 Cayley graphs and digraphs 44 3.4 The Doyle-Holt Graph 47 3.5 Non-Cayley vertex-transitive graphs 48 3.6 Coset graphs and Sabidussi’s Theorem 49 3.7 Double coset graphs and semisymmetric graphs 51 vii
- 14. viii Contents 3.8 Hamiltonicity 53 3.9 Characters of abelian groups and Cayley graphs 55 3.10 Growth rates 56 3.11 Exercises 58 3.12 Notes and guide to references 62 4 Orbital Graphs and Strongly Regular Graphs 64 4.1 Definitions and basic properties 64 4.2 Rank 3 groups 68 4.3 Strongly regular graphs 69 4.4 The Integrality Condition 70 4.5 Moore graphs 73 4.6 Exercises 75 4.7 Notes and guide to references 77 5 Graphical Regular Representations and Pseudosimilarity 79 5.1 Elementary results 79 5.2 Abelian groups 80 5.3 Pseudosimilarity 81 5.4 Some basic results 82 5.5 Several pairs of pseudosimilar vertices 84 5.6 Several pairs of pseudosimilar edges 85 5.7 Large sets of mutually pseudosimilar vertices 86 5.8 Exercises 88 5.9 Notes and guide to references 91 6 Products of Graphs 92 6.1 General products of graphs 93 6.2 Direct product 95 6.3 Cartesian product 97 6.4 More products 99 6.5 Stability and two-fold automorphisms 102 6.6 Additional remarks on graph products 105 6.7 Exercises 105 6.8 Notes and guide to references 108 7 Special Classes of Vertex-Transitive Graphs and Digraphs 109 7.1 Generalised Petersen graphs 110 7.2 Kneser graphs and odd graphs 114 7.3 Metacirculant graphs 115 7.4 The quasi-Cayley graphs and digraphs 117 7.5 Generalised Cayley graphs 119
- 15. Contents ix 7.6 Exercises 120 7.7 Notes and guide to references 122 8 The Reconstruction Conjectures 123 8.1 Definitions 124 8.2 Some basic results 126 8.3 Maximal planar graphs 132 8.4 Digraphs and degree-associated reconstruction 136 8.5 Exercises 138 8.6 Notes and guide to references 139 9 Reconstructing from Subdecks 140 9.1 The endvertex-deck 140 9.2 Reconstruction numbers 141 9.3 The characteristic polynomial deck 144 9.4 Exercises 147 9.5 Notes and guide to references 147 10 Counting Arguments in Vertex-Reconstruction 149 10.1 Kocay’s Lemma 149 10.2 Counting spanning subgraphs 151 10.3 The characteristic and the chromatic polynomials 154 10.4 Exercises 155 10.5 Notes and guide to references 156 11 Counting Arguments in Edge-Reconstruction 157 11.1 Definitions and notation 157 11.2 Homomorphisms of structures 159 11.3 Lovász’ and Nash-Williams’ Theorems 163 11.4 Extensions 166 11.5 Exercises 168 11.6 Notes and guide to references 169 References 171 List of Notations 185 Index of Terms and Definitions 187
- 17. Preface to the Second Edition In this second edition of our book we have tried to maintain the same structure as the first edition, namely a text which, although not providing an exhaustive coverage of graph symmetries and reconstruction, provides a detailed cover- age of some particular areas (generally motivated by our own research inter- est), which is not a haphazard collection of results but which presents a clear pathway through this thick forest. And our aim remains that of producing a text which can relatively quickly guide the reader to the point of being able to understand and carry out research in the topics which we cover. Among the additions in this edition we point out the use of the free com- puter programs GAP, GRAPE and Sage to construct and investigate some well- known graphs, including examples with properties like being semisymmetric, a topic which was treated in the first edition but for which examples are not easy to construct ‘by hand’. We have also updated some chapters with new results, improved the presentation and proofs of others, and introduced short treatments of topics such as character theory of abelian groups and their Cay- ley graphs to emphasise the connection between graph theory and other areas of mathematics. We have corrected a number of errors which we found in the first edition, and for this we would like to thank colleagues who have pointed out several of them, particularly Bill Kocay, Virgilio Pannone and Alex Scott. A special thanks goes to Russell Mizzi for help with overhauling Chapter 6, where we also introduce the new idea of two-fold isomorphisms, and to Leonard Soicher and Matan Zif-Av for several helpful tips regarding the use of GAP and GRAPE. The second author would like to thank the Politecnico di Milano for giving him the opportunity, by means of a sabbatical, to focus on the work needed xi
- 18. xii Preface to the Second Edition to complete the current edition of this book. He also thanks the University of Malta for its kind hospitality during this sabbatical. The authors will maintain a list of corrections and addenda at http://staff.um .edu.mt/josef.lauri. Josef Lauri Raffaele Scapellato
- 19. Preface to the First Edition This book arose out of lectures given by the first author to Masters students at the University of Malta and by the second author at the Università Cattolica di Brescia. This book is not intended to be an exhaustive coverage of graph theory. There are many excellent texts that do this, some of which are mentioned in the References. Rather, the intention is to provide the reader with a more in-depth coverage of some particular areas of graph theory. The choice of these areas has been largely governed by the research interests of the authors, and the flavour of the topics covered is predominantly algebraic, with emphasis on symmetry properties of graphs. Thus, standard topics such as the automorphism group of a graph, Frucht’s Theorem, Cayley graphs and coset graphs, and orbital graphs are presented early on because they provide the background for most of the work presented in later chapters. Here, more specialised topics are tackled, such as graphical regular representations, pseudosimilarity, graph products, Hamiltonicity of Cayley graphs and special types of vertex-transitive graphs, including a brief treatment of the difficult topic of classifying vertex-transitive graphs. The last four chapters are devoted to the Reconstruction Problem, and even here greater emphasis is given to those results that are of a more algebraic character and involve the symmetry of graphs. A special chapter is devoted to graph products. Such operations are often used to provide new examples from existing ones but are seldom studied for their intrinsic value. Throughout we have tried to present results and proofs, many of which are not usually found in textbooks but have to be looked for in journal papers. Also, we have tried, where possible, to give a treatment of some of these topics that is different from the standard published material (for example, the chapter on graph products and much of the work on reconstruction). xiii
- 20. xiv Preface to the First Edition Although the prerequisites for reading this book are quite modest (exposure to a first course in graph theory and some discrete mathematics, and elemen- tary knowledge about permutation groups and some linear algebra), it was our intention when preparing this book that a student who has mastered its con- tents would be in a good position to understand the current state of research in most of the specialised topics covered, would be able to read with profit journal papers in these areas, and would hopefully have his or her interest sufficiently aroused to consider carrying out research in one of these areas of graph theory. We would finally like to thank Professor Caroline Series for showing an interest in this book when it was still in an early draft form and the staff at Cambridge University Press for their help and encouragement, especially Roger Astley, Senior Editor, Mathematical Sciences, and, for technical help with L ATEX, Alison Woollatt, who, with a short style file, solved problems that would have baffled us for ages. Thanks are also due to Elise Oranges, who edited this book thoroughly and pointed out several corrections. The first author would also like to thank the Academic Work Resources Fund Committee and the Computing Services Centre of the University of Malta, the first for some financial help while writing this book and the second for technical assistance. He also thanks his M.Sc. students at the University of Malta, who worked through draft chapters of this book and whose comments and criticism helped to improve the final product. Josef Lauri Raffaele Scapellato
- 21. 1 Graphs and Groups: Preliminaries 1.1 Graphs and digraphs In these chapters a graph G = (V(G), E(G)) will consist of two disjoint sets: a nonempty set V = V(G) whose elements will be called vertices and a set E = E(G) whose elements, called edges, will be unordered pairs of distinct elements of V. Unless explicitly stated otherwise, the set of vertices will always be finite. An edge, {u, v}, u, v ∈ V, is also denoted by uv. Sometimes E is allowed to be a multiset, that is, the same edge can be repeated more than once in E. Such edges are called multiple edges. Also, edges uu consisting of a pair of repeated vertices are sometimes allowed; such edges are called loops. But unless otherwise stated, it will always be assumed that a graph does not have loops or multiple edges. The complement of the graph G, denoted by G, has the same vertex-set as G, but two distinct vertices are adjacent in the complement if and only if they are not adjacent in G. The degree of a vertex v, denoted by deg(v), is the number of edges in E(G) to which v belongs. A vertex of degree k is sometimes said to be a k-vertex. Two vertices belonging to the same edge are said to be adjacent, while a vertex and an edge to which it belongs are said to be incident. A loop incident to a vertex v contributes a value of 2 to deg(v). A graph is said to be regular if all of its vertices have the same degree. A regular graph with degree equal to 3 is sometimes called cubic. The minimum and maximum degrees of G are denoted by δ = δ(G) and = (G), respectively. In general, given any two sets A, B, then A−B will denote their set-theoretical difference, that is, the set consisting of all of the elements that are in A but not in B. Also, a set containing k elements is often said to be a k-set. If S is a set of vertices of a graph G, then G−S will denote the graph obtained by removing S from V(G) and removing from E(G) all edges incident to some vertex in S. If F is a set of edges of G, then G − F will denote the graph whose 1
- 22. 2 Graphs and Groups: Preliminaries vertex-set is V(G) and whose edge-set is E(G) − F. If S = {u} and F = {e}, we shall, for short, denote G − S and G − F by G − u and G − e, respectively. If S is a subset of the vertices of G, then G[S] will denote the subgraph of G induced by S, that is, the subgraph consisting of the vertices in S and all of the edges joining pairs of vertices from S. An important modification of the foregoing definition of a graph gives what is called a directed graph, or digraph for short. In a digraph D = (V(D), A(D)) the set A = A(D) consists of ordered pairs of vertices from V = V(D) and its elements are called arcs. Again, an arc (u, v) is sometimes denoted by uv when it is clear from the context whether we are referring to an arc or an edge. The arc uv is said to be incident to v and incident from u; the vertex u is said to be adjacent to v whereas v is adjacent from u. The number of arcs incident from a vertex v is called its out-degree, denoted by degout(v), while the number of arcs incident to v is called its in-degree and is denoted by degin(v). A digraph is said to be regular if all of its vertices have the same out-degree or, equivalently, the same in-degree. Sometimes, when we need to emphasise the fact that a graph is not directed, we say that it is undirected. The number of vertices of a graph G or digraph D is called its order and is generally denoted by n = n(G) or n = n(D), while the number of edges or arcs is called its size and is denoted by m = m(G) or m = m(D). A sequence of distinct vertices of a graph, v1, v2, . . . , vk+1, and edges e1, e2, . . . , ek such that each edge ei = vivi+1 is called a path. If we allow v1 and vk+1, and only those, to be the same vertex, then we get what is called a cycle. The length of a path or a cycle in G is the number of edges in the path or cycle. A path of length k is denoted by Pk+1 while a cycle of length k is denoted by Ck. The distance between two vertices u, v in a connected graph G, denoted by d(u, v), is the length of the shortest path joining u and v. The diameter of G is the maximum value attained by d(u, v) as u, v run over V(G), and the girth is the length of the shortest cycle. In these definitions, if we are dealing with a digraph and the ei = vivi+1 are arcs, then the path or cycle is called a directed path or directed cycle, respectively. Given a digraph D, the underlying graph of D is the graph obtained from D by considering each pair in A(D) to be an unordered pair. Given a graph G, the digraph ← → G is obtained from G by replacing each edge in E(G) by a pair of oppositely directed arcs. This way, a graph can always be seen as a special case of a digraph. We adopt the usual convention of representing graphs and digraphs by draw- ings in which each vertex is shown by a dot, each edge by a curve joining the
- 23. 1.2 Groups 3 corresponding pair of dots and each arc (u, v) by a curve with an arrowhead pointing in the direction from u to v. A number of definitions on graphs and digraphs will be given as they are required. However, several standard graph theoretic terms will be used but not defined in these chapters; these can be found in any of the references [257] or [259]. 1.2 Groups A permutation group will be a pair (, Y) where Y is a finite set and is a subgroup of the symmetric group SY, that is, the group of all permutations of Y. The stabiliser of an element y ∈ Y under the action of is denoted by y while the orbit of y is denoted by (y). The Orbit-Stabiliser Theorem states that, for any element y ∈ Y, || = |(y)| · |y|. If the elements of Y are all in one orbit, then (, Y) is said to be a transitive permutation group and is said to act transitively on Y. The permutation group is said to act regularly on Y if it acts transitively and the stabiliser of any element of Y is trivial. By the Orbit-Stabiliser Theorem, this is equivalent to saying that acts transitively on Y and || = |Y|. Also, acts regularly on Y is equivalent to saying that, for any y1, y2 ∈ Y, there exists exactly one α ∈ such that α(y1) = y2. One important regular action of a permutation group arises as follows. Let be any group, let Y = and, for any α ∈ , let λα be the permutation of Y defined by λα(β) = αβ. Let L() be the set of all permutations λα for all α ∈ . Then (L(), Y) defines a permutation group acting regularly on Y. This is called the left regular representation of the group on itself. One can similarly consider the right regular representation of the group on itself, and this is denoted by (R(), Y). The following is an important generalisation of the previous definitions. If is a group and H ≤ , let Y = /H be the set of left cosets of H in . For any α ∈ , let λH α be a permutation on Y defined by λH α (βH) = αβH. Let LH() be the set of all λH α for all α ∈ . Then (LH(), Y) defines a permutation group that reduces to the left regular representation of if H = {1}. Two permutation groups (1, Y1), (2, Y2) are said to be equivalent, denoted by (1, Y1) ≡ (2, Y2), if there exists a bijective isomorphism φ : 1 → 2 and a bijection f : Y1 → Y2 such that, for all y ∈ Y1 and for all α ∈ 1, f(α(x)) = φ(α)(f(x)).
- 24. 4 Graphs and Groups: Preliminaries Figure 1.1. Aut(G), Aut(H) are isomorphic but not equivalent In this case we also say that the action of 1 on Y1 is equivalent to the action of 2 on Y2, and sometimes we denote this simply by 1 = 2, when the two sets on which the groups are acting is clear from the context. Figure 1.1 shows a simple example of two graphs whose automorphism groups (to be defined later in this chapter) are isomorphic as abstract groups but clearly not equivalent as permutation groups since the sets (of vertices) on which they act are not equal. (See also Exercise 1.7.) Note in particular that, if (1, Y1) ≡ (2, Y2), then apart from 1 2 as abstract groups, and |Y1| = |Y2|, the cycle structure of the permutations of 1 on Y1 must be the same as those of 2 on Y2. However, the converse is not true; that is, 1 and 2 could be isomorphic and the cycle structures of their respective actions could be the same, but (1, Y1) might not be equivalent to (2, Y2) (see Exercise 1.9). If (, Y) is a permutation group acting on Y and Y is a union of orbits of Y, then we can talk about the action of restricted to Y , that is, the permutation group (, Y ) where, for α ∈ and y ∈ Y , α(y ) is the same as in (, Y). When Y is a union of orbits we also say that it is invariant under the action of because in this case α(y ) ∈ Y for all α ∈ and y ∈ Y . Also, ( , Y ) is said to be a subpermutation group of (, Y) if ≤ and Y is a union of orbits of acting on Y. The following is a useful well-known result on permutation groups whose proof is not difficult and is left as an exercise (see Exercise 1.10). Theorem 1.1 Let (, Y) be a permutation group acting transitively on Y. Let y ∈ Y, let H = y be the stabiliser of y and let W be /H, the set of left cosets of H in . Then (, Y) is equivalent to (LH(), W). If (, Y) is not transitive, and O is the orbit containing y, then (LH(), W) is equivalent to the action of on Y restricted to O. In the context of groups and graphs we shall need the very important idea of a group acting on pairs of elements of a set. Thus, let (, Y) be a permutation
- 25. 1.2 Groups 5 group acting on the set Y. By (, Y × Y) we shall mean the action on ordered pairs of Y induced by as follows: If α ∈ and x, y ∈ Y, then α((x, y)) = (α(x), α(y)). Similarly, by (, Y 2 ) we shall mean the action on unordered pairs of distinct elements of Y induced by α({x, y}) = {α(x), α(y)}. These ideas will be developed further in a later chapter. In later chapters we shall also need the notions of k-transitivity and primitiv- ity of a permutation group. In order to study permutation groups in more detail one has to dig deeper into the concept of transitivity. Suppose, for example, that Y is the set {1, 2, 3, 4, 5} and is the group generated by the permutation α = (1 2 3 4 5). Then clearly the permutation group (, Y) is transitive because for any i, j ∈ Y there is some power of α which maps i into j. But there is no power of α which, say, simultaneously maps 1 into 5 and 2 into 3. That is, not every ordered pair of distinct elements of Y can be mapped by a permutation in into any other given ordered pair of distinct elements. We therefore say that the permutation group (G, Y) is not 2-transitive. More generally, a permutation group (, Y) is said to be k-transitive if, given any two k-tuples (x1, x2, . . . , xk) and (y1, y2, . . . , yk) of distinct elements of Y, then there is an α ∈ such that (α(x1), α(x2), . . . , α(xk)) = (y1, y2, . . . , yk). Thus, a transitive permutation group is 1-transitive. Also, (, Y) is said to be k-homogeneous if, for any two k-subsets A, B of Y, there is an α ∈ such that α(A) = B, where α(A) = {α(a) : a ∈ A}. Finally, let (, Y) be transitive and suppose that R is an equivalence relation on Y, and let the equivalence classes of Y under R be Y1, Y2, . . . , Yr. Then (, Y) is said to be compatible with R if, for any α ∈ and any equiv- alence class Yi, the set α(Yi) is also an equivalence class. For example, if Y = {1, 2, 3, 4} and is the group generated by the permutation (1 2 3 4), then (, Y) is compatible with the relation whose equivalence classes are {1, 3} and {2, 4}. Any permutation group is clearly compatible with the trivial equivalence relations on Y, namely, those in which either all of Y is an equivalence class or when each singleton set is an equivalence class. If these are the only equiva- lence relations with which (, Y) is compatible, then the permutation group is said to be primitive. Otherwise it is imprimitive.
- 26. 6 Graphs and Groups: Preliminaries If (, Y) is imprimitive and R is a nontrivial equivalence relation on Y with which the permutation group is compatible, then the equivalence classes of R are called imprimitivity blocks and their set Y/R is an imprimitivity block system for the permutation group (, Y). It is an easy exercise (see Exercise 1.14) to show that a 2-transitive permu- tation group is primitive. We shall also need some elementary ideas on the presentation of a group in terms of generators and relations. Let be a group and let X ⊆ . A word in X is a product of a finite number of terms, each of which is an element of X or an inverse of an element of X. The set X is said to generate if every element in can be written as a word in X; in this case the elements of X are said to be generators of . A relation in X is an equality between two words in X. By taking inverses, any relation can be written in the form w = 1, where w is some word in X. If X generates and every relation in can be deduced from one of the relations w1 = 1, w2 = 1, . . . in X, then we write = X|w1 = 1, w2 = 1, . . . . This is called a presentation of in terms of generators and relations. The group is said to be finitely generated (respectively, finitely related) if |X| (respectively, the number of relations) is finite; it is called finitely presented, or we say that it has a finite presentation, if it is both finitely generated and finitely related. It is clear that every finite group has a finite presentation (although the con- verse is false). Simply take X = and, as relations, take all expressions of the form αiαj = αk for all αi, αj ∈ . In other words, the multiplication table of serves as the defining relations. It is well to point out that removing relations from a presentation of a group in general gives a larger group, the extreme case being that of the free group which has only generators and no relations. The simplest free group is the infinite cyclic group that has the presentation α with just one generator and no defining relation, whereas the cyclic group of order n has the presentation α|αn = 1 ; this group is denoted by Zn. The group with presentation α, β
- 27. 1.3 Graphs and groups 7 is the infinite free group on two elements. The dihedral group of degree n is denoted by Dn. It has order 2n and also has a presentation with two generators: α, β|α2 = 1, βn = 1, α−1 βα = β−1 . Determining a group from a given presentation is not an easy problem. The reader who doubts this can try to show that the presentations α, β : αβ2 = β3 α, βα2 = α3 β and α, β, γ : α3 = β3 = γ 3 = 1, αγ = γ α−1 , αβα−1 = βγβ−1 both give the trivial group. We shall of course make a very simple use of stan- dard group presentations where these difficulties do not arise. The book [159] is a standard reference for advanced work on group presentations. The reader is referred to [147, 222] for any terms and concepts on group theory that are used but not defined in these chapters and, in particular, to [49, 62] for more information on permutation groups. 1.3 Graphs and groups Let G, G be two graphs. A bijection α : V(G) → V(G ) is called an isomor- phism if {u, v} ∈ E(G) ⇔ {α(u), α(v)} ∈ E(G ). The graphs G, G are, in this case, said to be isomorphic, and this is denoted by G G . Similarly, if D, D are digraphs, then a bijection α : V(D) → V(D ) is called an isomorphism if (u, v) ∈ A(D) ⇔ (α(u), α(v)) ∈ A(D ), and in this case the digraphs D, D are also said to be isomorphic, and again this is denoted by D D . If the two graphs, or digraphs, in this definition are the same, then α is said to be an automorphism of G or of D. The set of automorphisms of a graph or a digraph is a group under composition of functions, and it is denoted by Aut(G) or Aut(D). Note that an automorphism α of G is an element of SV(G), although it is its induced action on E(G) that determines whether α is an automorphism. This fact, although clear from the definition of automorphism, is worth emphasising when beginning to study automorphisms of graphs.
- 28. 8 Graphs and Groups: Preliminaries Figure 1.2. No automorphism permutes the edges as (12 23 34) For example, for the graph in Figure 1.2, the permutation of edges given by (12 23 34) is not induced by any permutation of the vertex-set {1, 2, 3, 4}. The only automorphisms for this graph are the identity and the permutation (14)(23), which induces the permutation (12 34)(23) of the edges in the graph. The question of edge permutations not induced by vertex permutations will be considered in some more detail later in this chapter. The process of obtaining a permutation group from a digraph can be reversed in a very natural manner. Suppose that (, Y) is a group of permutations acting on a set Y. Let A be a union of orbits of (, Y×Y). Clearly, the digraph D whose vertex-set is Y and whose arc-set is A has as a subgroup of its automorphism group. It might, however, happen that Aut(G) is larger than . Moreover, if the pairs in A are such that, for every (u, v) ∈ A, (v, u) is also in A, then replacing every opposite pair of arcs of D by a single edge gives a graph G such that ⊆ Aut(G). This and other ways of constructing graphs or digraphs admitting a given group of permutations will be studied in more detail in Chapter 4. Certain facts about automorphisms of graphs and digraphs are very easy to prove and are therefore left as exercises: (i) Aut(G) = Aut(G); (ii) Aut(G) = SV(G) if and only if G or G is Kn, the complete graph on n vertices; (iii) Aut(Cn) = Dn. Also, let α be an automorphism of G and u, v vertices of G. Then, (iv) deg(u) = deg(α(u)); (v) G − u G − α(u); (vi) d(u, v) = d(α(u), α(v)), where d(u, v) is the distance between u and v. Also, if u is a vertex in a digraph D and α is an automorphism of D, then (vii) degin(u) = degin(α(u)) and degout(u) = degout(α(u)). If u and v are vertices in a graph G and there is an automorphism α of G such that α(u) = v, then u and v are said to be similar. If G − u G − v, then u and v are said to be removal-similar. Property (v) tells us that if two vertices are similar, then they are removal-similar. The converse of this is, however,
- 29. 1.3 Graphs and groups 9 false, as can be seen from the graph shown in Figure 1.3. Here, the vertices u, v are removal-similar but not similar. Such vertices are called pseudosimi- lar. Similar, removal-similar and pseudosimilar edges are analogously defined: Two edges ab, cd of G are similar if there is an automorphism α of G such that α(a)α(b) = cd. We shall be studying pseudosimilarity in more detail in Chapter 5. Sometimes we ask questions of this type: how many graphs (possibly of some fixed order n) are there? The answer to this question depends heavily on how we consider two graphs to be different. In general, if the order of a graph G is n, we can think of its vertices as being labelled with the integers {1, 2, . . . , n}. Two graphs G and H of order n so labelled are called identical or equal as labelled graphs (written G = H) if ij ∈ E(G) ⇔ ij ∈ E(H). (Compare this definition with that of isomorphic graphs.) Obviously, identical graphs are isomorphic, but the converse is not true. For example, the graphs in Figure 1.4 are isomorphic but not identical. Counting nonisomorphic graphs is, in general, much more difficult than counting nonidentical graphs. For example, there are four nonisomorphic graphs on three vertices but eight nonidentical ones. These are shown in Figures 1.5 and 1.6, respectively. Figure 1.3. A pair of pseudosimilar vertices Figure 1.4. Isomorphic but nonidentical graphs Figure 1.5. The four nonisomorphic graphs of order 3
- 30. 10 Graphs and Groups: Preliminaries Figure 1.6. The eight nonidentical graphs of order 3 Counting nonisomorphic graphs involves consideration of group symme- tries. For more on this the reader is referred to [103]. 1.4 Edge-automorphisms and line-graphs Although we shall be dealing mostly with Aut(G) and its realisation as the permutation group (Aut(G), V(G)), let us briefly look at other related groups associated with G. In this section we shall assume that G is a nontrivial graph, that is, its edge-set is nonempty. An edge-automorphism of a graph G is a bijection θ on E(G) such that two edges e, f are adjacent in G if and only if θ(e), θ(f) are also adjacent in G. The set of all edge-automorphisms of G is a group under composition of functions, and it is denoted by Aut1(G). The concept of edge-automorphisms can perhaps be best understood within the context of line-graphs. The line-graph L(G) of a graph G is defined as the graph whose vertex-set is E(G) and in which two vertices are adjacent if and only if the corresponding edges are adjacent in G. An automorphism of L(G) is clearly an edge-automorphism of G and (Aut1(G), E(G)) is equivalent to (Aut(L(G), V(L(G))). In this section we shall give the exact relationship between Aut1(G) and Aut(G), that is, between the automorphism groups of G and L(G). As we described earlier, any automorphism α of G naturally induces a bijec- tion α̂ on E(G) defined by α̂(uv) = α(u)α(v). It is an important (and easy to
- 31. 1.4 Edge-automorphisms and line-graphs 11 verify) property of α̂ that two edges e1, e2 are adjacent if and only if α̂(e1), α̂(e2) are adjacent, that is, if and only if α̂ is an edge-automorphism. For this reason α̂ is called an induced edge-automorphism of G. The set of all induced edge-automorphisms of G is denoted by Aut∗ (G), and it is easy to check that this is a subgroup of Aut1(G) under composition of functions. Now, it seems natural to expect that Aut(G) and Aut∗ (G) are iso- morphic. However, it can happen that two different automorphisms of G induce the same edge-automorphism. For example, let G = K2. Then |Aut(G)| = 2 but |Aut∗(G)| = 1. Also, suppose that G contains isolated vertices. Then any automorphism of G that permutes the isolated vertices and leaves all of the oth- ers fixed induces the trivial edge-automorphism. The following theorem says that these are basically the only situations when Aut(G) Aut∗ (G). Theorem 1.2 Let G be a nontrivial graph. Then Aut(G) Aut∗(G) if and only if G has at most one isolated vertex and K2 is not a component. Proof Clearly, the mapping α → α̂ is a homomorphism from Aut(G) onto Aut∗(G) because α̂.β̂(uv) = α.β(u)α.β(v) = αβ(uv). We must therefore show that the kernel of this mapping is trivial if and only if G has at most one isolated vertex and K2 is not a component. Suppose first that G has two isolated vertices u, v or K2 as a component with vertices u, v. Then the permutation α that transposes u and v and fixes all of the other vertices is a nontrivial automorphism of G, but α̂ is the identity. Therefore the kernel is not trivial. Conversely, suppose that G does not contain K2 as a component nor its com- plement. If Aut(G) is trivial, then so is Aut∗(G). Therefore, let α be a nontrivial element of Aut(G), and let α(u) = v = u. Then deg(u) = deg(v) = 0 (other- wise u, v would be a pair of isolated vertices). We consider two cases. Case 1: u, v adjacent. Let e be the edge uv. Then deg(u) = deg(v) 1 (otherwise the two vertices u, v would form a component K2). Therefore, there exists an edge f = e incident to u (but not to v, since the graph is simple). But α̂(f) must be incident to v (since α(u) = v), that is, α̂(f) = f, and hence α̂ is not trivial. Case 2: u, v not adjacent. Let e be an edge incident to u. Again, e is not incident to v but α̂(e) is. Therefore α̂ is again nontrivial. The next natural question to ask is whether there can be edge-automorphisms of G that are not induced by automorphisms, that is, whether Aut∗(G) can be a strict subgroup of Aut1(G). This situation can very well happen, although, as we shall see, such cases are quite rare.
- 32. 12 Graphs and Groups: Preliminaries Figure 1.7. Graphs with edge-isomorphisms not induced by isomorphisms Before proceeding let us first extend the idea of edge-automorphisms on the edge-set of a graph to that of edge-isomorphisms between edge-sets of different graphs. Let G, G be two nontrivial graphs. A bijection θ : E(G) → E(G ) is an edge-isomorphism if e, f adjacent in G ⇔ θ(e), θ(f) adjacent in G . Two graphs are said to be edge-isomorphic if there is an edge-isomorphism between their edge-sets. The graphs W1, W2 in Figure 1.7 are edge-isomorphic, although they are not isomorphic. That is, there is an edge-isomorphism between their edge- sets that cannot be induced by an isomorphism between their vertex-sets. This means that their line-graphs are isomorphic even though the two graphs are not themselves isomorphic. Also, each of the graphs W3, W4, W5 in the same figure has edge- automorphisms that are not induced by automorphisms. That is, the group Aut∗ (Wi) is a strict subgroup of Aut1(Wi). In other words, Aut(L(Wi)) is larger than Aut(Wi). The following theorem of Whitney [258] says that these are essentially the only cases when edge-isomorphisms that are not induced by isomorphisms can arise. We give the statement of the theorem without proof, which, although not deep or difficult, would lengthen this introductory chapter without adding significant new insights. Theorem 1.3 (Whitney) Let G, G be connected graphs different from the five graphs in Figure 1.7. Let θ : E(G) → E(G ) be an edge-isomorphism. Then θ is induced by an isomorphism from G to G .
- 33. 1.5 A word on issues of computational complexity 13 Whitney’s Theorem and Theorem 1.2 together therefore give the following corollary. Corollary 1.4 Let G be a nontrivial graph. Then Aut1(G) = Aut∗(G) if and only if both of these conditions hold: (i) not both W1, W2 are components of G; (ii) none of Wi, i = 3, 4, 5 is components of G. Moreover, Aut1(G) Aut(G) (that is, Aut(L(G)) Aut(G)) if and only if (i) and (ii) hold and G has at most one isolated vertex and K2 is not a component of G. 1.5 A word on issues of computational complexity Although in this book we shall not concern ourselves with issues of computa- tional complexity, it is perhaps worthwhile to say a few words in this regard here in order to put matters into a better perspective. A student reading the def- initions of isomorphic graphs and automorphisms might think that it is an easy matter to determine in general whether two given graphs are isomorphic or to compute the automorphism group of a graph. In fact, this is far from being the case, and these problems are very hard to crack in practice, at least as far as present knowledge goes. In general, one considers that an efficient algorithm exists for finding a solu- tion to a problem (for example, finding a nontrivial automorphism of a given graph) if there is a general algorithm such that the number of operations that it takes to solve the problem is a polynomial function of the size of the input (say, the number of vertices in the graph); one says that the algorithm solves the problem in polynomial time. Of course, several terms in the previous sentence need exact definitions, but we shall here take an intuitive approach and refer the reader to [36] or [82] for the exact details on computational complexity. Those problems for which an efficient (polynomial-time) algorithm exists form the class denoted by P (which stands for ‘polynomial’). However, there are several problems for which it is not known whether an efficient algorithm does exist. In order to tackle this question of computational intractibility, two important ideas have been developed. Firstly, the class NP (which stands for ‘nondeterministic polynomial’) is defined. Roughly (again we refer the reader to the textbooks cited earlier for the exact details) this class contains all of those problems for which, given a candidate solution, one can verify in polynomial time that it is in fact a correct
- 34. 14 Graphs and Groups: Preliminaries solution. For example, the problem of determining whether a graph has a non- trivial automorphism is in NP, since, given such a permutation of the vertices, it is easy to determine in polynomial time that it is an automorphism. Now the main question in computational complexity is whether P = NP (clearly P ⊆ NP), and to tackle this question another important idea is intro- duced. Given two problems A and B, one says that A is (polynomially) reducible to B if, given an algorithm for solving B, it can be transformed in polynomial time into an algorithm for solving A. Reducibility therefore introduces a hier- archy between problems for, if A is reducible to B, then, in a sense, A cannot be more difficult to solve (computationally) than B. In particular, if there is an efficient algorithm for solving B, then there is also an efficient algorithm for solving A. Now, the question of reducibility took on special significance by the discov- ery that in the class NP there are problems, called NP-complete, to which any other problem in NP is reducible. In other words, if an efficient algorithm can be found for any NP-complete problem, then all problems in NP would have an efficient algorithm to solve them, and P would be equal to NP. Now, it is not known whether the problem of determining if two graphs are isomorphic, which lies clearly in NP, is NP-complete. In fact, if it turns out that P is not equal to NP, then there is evidence to suggest that the problem of graph isomorphism might lie strictly between the classes P and NP. What all this means in practice is that, as far as present knowledge goes, no general algorithm can determine in a guaranteed reasonable time whether two graphs are isomorphic, or whether a given graph has a nontrivial automorphism (these two problems are closely related [126]). It is known that for special types of graphs (for example, trees, planar graphs and graphs with bounded degree) an efficient algorithm does exist. Computer packages can also help one to solve these problems, certainly more efficiently than an attempt ‘by hand’ for large graphs, although their time performance is not guaranteed (by what we said earlier). For example, the soft- ware package MathematicaTM has a combinatorics extension that, amongst other things, finds graph automorphisms and isomorphisms. A more specialised package, and one that is freely available from www.combinatorialmath.org.ca/gg/index.html is Groups Graphs [131], developed by Bill Kocay. This package contains several combinatorial routines related to graphs, digraphs, combinatorial designs and their automorphism groups and also embeddings of graphs on some sur- faces and a graph isomorphism algorithm. It is easy to use, and it has a pleasant graphical user interface. It is also very useful simply for drawing diagrams of
- 35. 1.6 Exercises 15 graphs. Although originally written for MacintoshTM computers, a version for the unix-based Haiku operating system is in preparation, and this version will contain several new features. An important computer algebra package, which is also freely available, is the system GAP [243]. This package performs very sophisticated routines in discrete abstract algebra, in particular routines on permutation groups. It incor- porates a number of extensions, one of which, GRAPE [235], deals specifically with graphs, including their automorphisms and isomorphisms. The computer package Sage [227] is an open-source competitor to systems like MapleTM, MathematicaTM and MatlabTM. It incorporates several open- source mathematical software like GAP and R, and it can be run via Sage- MathCloud without the need of installing the system on one’s computer. It has an excellent library of functions for doing graph theory. In this book we shall present some constructions using GAP and Sage. Finally, it should be mentioned that it is generally accepted that the best package to tackle graph isomorphisms is nauty [181], developed by Brendan McKay. In fact, the system GRAPE invokes nauty when computing automor- phisms or isomorphisms. 1.6 Exercises 1.1 Draw all twenty nonidentical graphs with vertex-set {1, 2, 3, 4} that have three edges. How many of them are nonisomorphic? In general, how many nonidentical graphs on n vertices and m edges are there? How many are there on n vertices? 1.2 Let G be the graph in Figure 1.8. How many nonidentical labellings does G have using the labels {1, 2, . . . , 6} on its vertices? In general, how many nonidentical labellings does a graph G on n vertices have using the labels {1, 2, . . . , n} on its vertices? 1.3 Show that if G is self-complementary (that is, G G), then n ≡ 0 mod 4 or n ≡ 1 mod 4. Determine all self-complementary graphs on five vertices. 1.4 A well-known result due to Cayley says that the number of nonidentical trees on n vertices is nn−2. Verify this for n = 4. Look up one of the several proofs of this result. Figure 1.8. How many distinct labellings does this graph have?
- 36. 16 Graphs and Groups: Preliminaries The points 1, 2, . . . , n are drawn in a plane. A random tree is drawn joining these points, with all possible spanning trees being equally likely. Let pn be the probability that 1 is an endvertex of the tree. Show that limn→∞ pn = 1/e. 1.5 Find a graph with a pair of pseudosimilar edges. 1.6 Let Y = {1, 2, 3, 4} and let be the group acting on Y generated by the permu- tation (1 2 3 4). Construct a digraph D whose vertex-set is Y and whose arc-set is the orbit of the arc (1, 2) under the action of the permutation group (, Y × Y). Is the whole of Aut(D)? Can a graph be obtained by taking the orbit of some other arc or a union of orbits? Will always be the whole of Aut(D)? 1.7 Let P be a rectangular plate and Q a plate in the form of a rhombus. Show that the groups of symmetry of P and Q are isomorphic as abstract groups but not equivalent considered as permutation groups of the four vertices of P and Q. Show that the same situation arises with the following two graphs: the cycle on four vertices with an extra multiple edge and the complete graph K4 with an edge deleted. 1.8 Show that the dihedral group Dn can be presented as α, β|α2 = β2 = (αβ)n = 1 . 1.9 Let 1 be the abelian group defined by the presentation α, β, γ |α3 = β3 = γ 3 = 1, [α, β] = [α, γ ] = [β, γ ] = 1 , where [α, β] = α−1β−1αβ is the commutator of α and β, and let 2 be the group defined by the presentation α, β, γ |α3 = β3 = γ 3 = 1, [β, α] = γ , [α, γ ] = [β, γ ] = 1 . Show that although 1 and 2 are not isomorphic as abstract groups, and there- fore the two permutation groups (L(1), 1) and (L(2), 2) are not equivalent, still the cycle structures of the permutations of L(1) acting on 1 are the same as those of the permutations of L(2) acting on 2. Give an example of two permutation groups whose permutations have the same cycle structures and which are isomorphic as abstract groups but are still not equivalent as permutation groups. 1.10 Prove Theorem 1.1. 1.11 This exercise is intended to illustrate Theorem 1.1. Consider the action of the alternating group A4 on the set X = {1, 2, 3, 4}. Let H be the stabiliser of the element 1 under this action. Confirm that the left action of A4 on the left cosets of H is equivalent to the action of A4 on X. 1.12 Let be a group of permutations acting on a set Y and let y, x be two elements of Y that are in the same orbit under this action. Let α, β ∈ be two permutations such that α(y) = β(y) = x. Prove that αy = βy. Prove also that if x is in the same orbit as y and γ is a permutation such that γ (y) = x and γ y = αy, then x = x. 1.13 Show that if the abelian permutation group acts transitively on Y, then its action on Y is regular. 1.14 (a) Show that a 2-transitive permutation group is primitive. (b) Show that if Aut(G), for a graph G, is 2-transitive, then either G or its com- plement is a complete graph.
- 37. 1.7 Notes and guide to references 17 (c) Suppose that the transitive permutation group (, Y), with |Y| finite, is imprim- itive. Show that the blocks of an imprimitivity block system have equal size. 1.15 Let G be a vertex-transitive graph whose automorphism group acts impri- mitively on V(G). Show that the subgraphs of G induced by the blocks of an imprimitivity block system are all isomorphic. Suppose that each such subgraph is replaced by its complement, leaving the other edges intact. Let G be the resulting graph. Show that Aut(G ) = Aut(G). 1.16 The Petersen graph can be defined as follows. Let N = {1, 2, 3, 4, 5}, and let the vertices of the graph be all subsets of N of size 2 in which two vertices are adja- cent if the corresponding subsets are disjoint. Use this definition and GAP (with GRAPE) to construct the Petersen graph and to verify some of its properties. 1.7 Notes and guide to references One of the standard texts on graph theory has, for many years, been [97]. More recent books that give an excellent coverage of the subject are [28, 61, 257, 259]. The last reference is a short introduction that is quite sufficient back- ground for this book. Biggs’ book [24] is the standard text on algebraic graph theory, but the more recent [90] is also an excellent and up-to-date textbook on the subject. The book [94] contains a number of recent and specialised survey papers on various aspects of algebraic graph theory, particularly those dealing with graph symmetries. A proof of Whitney’s Theorem can be found in [22]. We shall need only the most elementary notions of group theory. The text [147] gives ample coverage for our purposes, while [222] provides a more complete treatment. Two excellent books devoted entirely to permutation groups are [49, 62]. Most of the results and definitions on permutation groups that we have given here and others that we shall need can be found in the first few chapters of these two books. For a full discussion of the terms on computational complexity that were introduced earlier rather intuitively, the reader is referred to the standard text- book [82] or the more recent [36]. The book [126] and the references it cites are suggested for those who are interested in the computational complexity of the graph isomorphism problem. Those who are particularly interested in some of the powerful algebraic techniques used to tackle this problem should look at the papers [106, 155]. For practical computations on a computer with per- mutation groups and graph automorphisms and isomorphisms in particular, the systems [131, 181, 235, 243] are recommended.
- 38. 2 Various Types of Graph Symmetry We shall see in this chapter that most graphs are asymmetric, that is, their automorphism group is trivial; in other words, it consists only of the identity permutation. The least number of vertices that an asymmetric graph can have is six, and the graph shown in Figure 2.1 is the smallest such graph in the sense that any other asymmetric graph on six vertices has more edges. As is to be expected, however, the most interesting relationships between groups and graphs arise when the graphs have a very high degree of symmetry, that is, a large automorphism group. One way to make more precise the idea of a large automorphism group is to require that it at least be transitive on the vertex-set or the edge-set of the graph. The weakest forms of symmetry to ask of a graph involve vertex-transitivity and edge-transitivity, which we define in the next section. 2.1 Transitivity Recall the definition of similar vertices from the previous chapter. We say that a graph G is vertex-transitive if any two vertices of G are similar, that is, if, for any u, v ∈ V(G), there is an automorphism α of G such that α(u) = v. In other words, G is vertex-transitive if all of the vertices of G are in the same orbit of the permutation group (Aut(G), V(G)). Figure 2.1. The smallest asymmetric graph 18
- 39. 2.1 Transitivity 19 Figure 2.2. A vertex-transitive graph that is not edge-transitive One can define edge-transitivity analogously. A graph G is edge-transitive if, given any two edges {a, b} and {c, d}, there exists an automorphism α such that α{a, b} = {c, d}, that is, {α(a), α(b)} = {c, d}. In other words, G is edge- transitive if any two of its edges are similar under the action of the permutation group (Aut∗(G), E(G)), that is, if the edges of G are all in one orbit under this action. Note that very often the word ‘transitive’ is used to refer to a graph, and in this case it is taken to mean ‘vertex-transitive’. Vertex-transitivity does not imply edge-transitivity, nor does the converse implication hold. Figure 2.2 shows a graph that is vertex-transitive but not edge-transitive. The complete bipartite graph Kp,q with p = q is a simple example of a graph that is edge-transitive but not vertex-transitive. The following well-known result does give a description of edge-transitive graphs that are not vertex- transitive. Theorem 2.1 Let G be a graph without isolated vertices and let H be a sub- group of Aut(G). Suppose that the action induced by H is transitive on the edges of G but not on its vertices. Then G is bipartite and the action of H on V(G) has two orbits that form the bipartition of V(G). Proof Let {u, v} be an edge of G. Let V1, V2 be the orbits under the action of H containing u and v, respectively. (We are not excluding, for the moment, the possibility that V1 = V2.) Let x be any other vertex of G. Since G does not have isolated vertices, there exists a vertex y adjacent to x; that is, {x, y} is an edge of G. But G is edge-transitive under the action of H; therefore the two edges are similar under this action. Hence x is similar to at least one of u or v, that is, x is in V1 or V2. Therefore V1 ∪ V2 = V(G). Now, V1, V2 must be disjoint; otherwise (since orbits form a partition) they are equal, and this would mean that the vertices of G are all in one orbit, giving that G is vertex-transitive under the action of H. Hence we now have that the action of H on V(G) has exactly two orbits, V1, V2.
- 40. 20 Various Types of Graph Symmetry Now let a, b be in the same orbit, say V1. It then follows that a, b are not adjacent. For suppose otherwise. Then the edge {u, v} is similar to the edge {a, b} under the action of H; therefore the vertex v is similar to one of a or b under this action, giving that v (which is in the orbit V2) is also in the orbit V1, which contains a and b. But this is impossible since V1 ∩ V2 = ∅. Hence, as required, we have that no two vertices from the same orbit can be adjacent. Corollary 2.2 If a graph G without isolated vertices is edge-transitive but not vertex-transitive, then it is bipartite and the action of Aut(G) on V(G) has two orbits that form the bipartition of V(G). Proof Take H = Aut(G) in the previous theorem. 2.1.1 Semisymmetric graphs The typical example of edge-transitive but not vertex-transitive graphs given earlier is the complete bipartite graph with a different number of vertices in the bipartition. These graphs are trivially not vertex-transitive because their vertices have different degrees. Although regular graphs do exist that are edge- transitive but not vertex-transitive, it is quite difficult to construct them. Such graphs are now called semisymmetric graphs and they were first studied by Folkman [77], who constructed the smallest possible semisymmetric graph having twenty vertices.1 One construction of the Folkman Graph is described in Exercise 2.5. Here we shall describe another well-known semisymmetric graph, the Gray Graph.2 For a long time nobody could find a smaller cubic semisymmetric graph, and eventually it was formally proved in [160] that it is the smallest cubic semisymmetric graph. We shall see a more systematic way of describing it in subsequent chapters. Here we follow Bouwer’s construction in [33]. Consider a cycle on 54 vertices numbered consecutively from 0 to 53. To form the Gray Graph G add the following edges to this cycle: {1, 42}, {2, 15}, {3, 28}, {4, 33}, {5, 44}, {6, 53}, {7, 48}, {8, 21}, {9, 32}, {10, 45}, {11, 24}, {12, 41}, {13, 20}, {14, 31}, {16, 35}, {17, 40}, {18, 49}, {19, 0}, {22, 51}, {23, 30}, {25, 38}, {26, 43}, {27, 52}, {29, 36}, {34, 47}, 1 More information about this graph, called the Folkman Graph, can be found on the MathWorld page http://mathworld. wolfram.com/FolkmanGraph.html 2 The Gray Graph is also featured on MathWorld at http://mathworld.wolfram.com/Gray Graph.html
- 41. 2.1 Transitivity 21 {37, 50}, {39, 46}. It is tedious but not difficult to check that the permutations α = (2 0 43)(3 53 43)(4 6 44)(7 45 33)(8 10 32)(11 31 21) = (12 14 20)(15 19 41)(16 18 40)(22 24 30)(25 29 51) = (26 28 52)(34 48 46)(35 49 39)(36 50 38) and β = (1 7 11 37 15 53 9 25 35)(2 6 10 38 16 0 8 24 36) = (3 5 45 39 17 19 24 23 29)(4 44 46 40 18 20 22 30 28) = (12 50 14 52 32 26 34 42 48)(13 51 31 27 33 43 47 41 49) are automorphisms of G. Note that the automorphism α fixes the vertex 1 and permutes cyclically its neighbours 2, 42 and 0. Thus, in order to show that the graph is edge-transitive it is sufficient to show that any odd-numbered vertex can be mapped into 1 by an automorphism of G. This can be done by appropriate products of α and β. For example, α4β maps vertex 53 to vertex 1. However, the graph is not vertex-transitive because from an odd-numbered vertex it is possible to have three different paths of length 4 joining the vertex to some other common vertex (for example, vertex 1 to vertex 5), but this is not possible from an even-numbered vertex. Another way to show that the Gray Graph is not vertex-transitive is to con- sider the distance sequences of its vertices [166]. The distance sequence of a vertex v is the vector (a0, a1, . . . , ar) where ai is the number of vertices at dis- tance i from v. In the case of the Gray Graph, the distance sequences of the vertices in the two colour classes are (1, 3, 6, 12, 12, 12, 8) and (1, 3, 6, 12, 16, 12, 4), respectively, therefore these vertices cannot be in the same orbit under the automorphism group of the graph. Although the Sage package has the Gray Graph already implemented, it is easy and instructive to show how to construct it following the specification given earlier. First one creates the vertex-set which will be the list of numbers from 1 to 54. Sage, like many computer languages such as Python, starts its lists from 0. Therefore a list of length n produced by the command range(55) would contain the numbers from 0 to 54. To start the list from 1 we have to define the list of vertices as vertices := range(1,55);
- 42. 22 Various Types of Graph Symmetry The graph is then constructed first using the command DiGraph so that we do not need to repeat every pair of adjacent vertices twice. This com- mands basically takes two parameters. The first parameter is the vertex-set of the graph to be constructed, and the second parameter is a boolean function (defined with the lambda construct) of two variables which compares all pos- sible pairs of the vertex-set and an edge is drawn between any pair of vertices for which the function returns True. dgray := DiGraph([vertices, lambda i, j: (i == mod[j + 1, 54]) or (i == 53 and j == 54) or (i == 1 and j == 42) or (i == 2 and j == 15) or (i == 3 and j == 28) or (i == 4 and j == 33) or (i == 5 and j == 44) or (i == 6 and j == 53) or (i == 7 and j == 48) or (i == 8 and j == 21) or (i == 9 and j == 32) or (i == 10 and j == 45) or (i == 11 and j == 24) or (i == 12 and j == 41) or (i == 13 and j == 20) or (i == 14 and j == 31) or (i == 16 and j == 35) or (i == 17 and j == 40) or (i == 18 and j == 49) or (i == 19 and j == 54 or (i == 22 and j == 51) or (i == 23 and j == 30) or (i == 25 and j == 38) or (i == 26 and j == 43) or (i == 27 and j == 52) or (i == 29 and j == 36) or (i == 34 and j == 47) or (i == 37 and j == 50) or (i == 39 and j == 46) ] ) This digraph is then changed into an undirected graph with the following command which changes every arc into an edge.
- 43. 2.1 Transitivity 23 gray = dgray.to_undirected() It is then easy to check that the aforementioned properties of the Gray Graph hold. For example, in order to check whether it is vertex-transitive we use the command gray.is_vertex_transitive() which returns False. The command gray.is_edge_transitive() returns True, as expected, while the command gray.is_regular() also returns True, confirming that the graph is semisymmetric. In fact, we could have reached the same conclusion with the command gray.is_semi_symmetric() which also returns True. Finally, one can check whether the graph constructed earlier is isomorphic to Sage’s inbuilt ‘GrayGraph’ using the command graphs.GrayGraph.is_isomorphic(gray) which, again, returns True. We shall have more to say about the Gray Graph in a later chapter when we shall describe it in a more algebraic fashion. Exercises 2.2 and 2.5 show that any semisymmetric graph must have even order and its degree must be less than |V(G)|/2. 2.1.2 Arc-transitive and 1 2 -arc-transitive graphs A stronger form of transitivity than either vertex- or edge-transitivity based on the edge-set of G can also be defined. If G has the property that, for any two edges {a, b}, {c, d}, there is an automorphism α such that α(a) = c and α(b) = d and also an automorphism β such that β(a) = d and β(b) = c, then G is said to be arc-transitive. We shall now derive a result of Tutte that gives a restriction on the degree of the vertices of a graph that is vertex-transitive and edge-transitive but not arc-transitive. One can think of arc-transitivity as follows. Given any graph G, construct the directed graph ← → G obtained from G by replacing each edge {a, b} by the pair of
- 44. 24 Various Types of Graph Symmetry arcs (a, b) and (b, a). Then clearly Aut(G) = Aut( ← → G ) and any automorphism α of G induces the natural action on arcs given by (a, b) → (α(a), α(b)). Then G is arc-transitive precisely if, given any two arcs in ← → G , there is an automorphism of ← → G mapping one arc into the other. This is a stronger form of transitivity than both vertex- and edge-transitivity because now, given any two edges on each of which an orientation is imposed, there is an automorphism mapping one edge into the other and preserving the given orientations. In fact, an arc-transitive graph is both vertex-transitive and edge-transitive. Lemma 2.3 Let H be a subgroup of Aut(G) such that, under the action of H, G is vertex-transitive and edge-transitive but not arc-transitive. Let t be an arc of ← → G and let D be the subdigraph of ← → G whose vertex-set is V( ← → G ) and whose arc-set is the orbit of t under the action of H. Then (i) for every edge {a, b} of G, D contains exactly one of the arcs (a, b) or (b, a); (ii) H ≤ Aut(D); (iii) D is vertex-transitive. Proof (i) Let t = (s1, s2). Because G is edge-transitive under the action of H, there is an α ∈ H such that α{s1, s2} = {a, b}. Therefore certainly one of (a, b) or (b, a) is an arc of D. Suppose that both are arcs of D. Then there is some β ∈ H such that β((a, b)) = (b, a). But given any edge {c, d} of G there is, by edge-transitivity, a γ ∈ H such that γ ((a, b)) equals (c, d) or (d, c). Suppose, without loss of generality, that γ ((a, b)) = (c, d). But then γβ((a, b)) = (d, c). Therefore, for any edge {c, d} of G, both arcs (c, d) and (d, c) are in the same orbit, that is, the action of H on G is arc-transitive, a contradiction. (ii) This follows because the arc-set of D is a full orbit of the permutation group (H, V(G) × V(G)). (iii) This follows because (H, V(G)) is transitive, V(D) = V(G) and H ≤ Aut(D). If we let H = Aut(G) in this lemma, then, in view of (i), if G is a vertex- transitive and edge-transitive graph that is not arc-transitive, it follows that the arc-set of ← → G is naturally partitioned into two orbits of equal size under the action of Aut(G), and none of the two orbits contains both an arc (a, b) and its inverse (b, a). In view of this, a graph that is vertex-transitive, edge-transitive but not arc-transitive is said to be 1 2 -arc-transitive.
- 45. 2.2 Asymmetric graphs 25 Figure 2.3. Relationship between different types of transitivity The relationship between these forms of transitivity is shown in Figure 2.3, where a line leading down from one property to another means that the first implies the second. Although 1 2 -arc-transitive graphs are not easy to find, they do exist (an exam- ple will be given in Chapter 3). The following well-known theorem of Tutte tells us that such a graph must have even degree. Theorem 2.4 (Tutte) Let H be a subgroup of Aut(G) such that, under the action of H, G is vertex-transitive and edge-transitive but not arc-transitive. Then the degree of G is even. In particular, a 1 2 -arc-transitive graph has even degree. Proof Let D be as in the previous lemma. By the third part of this lemma, all vertices of D have the same out-degree, say k. Now, k·|V(D)| = |A(D)| and, by the first part of the lemma, |A(D)| = |E(G)|. But if the common degree of the vertices of G is d, then, by the Handshaking Lemma, |E(G)| = d · |V(G)|/2 = d · |V(D)|/2. Therefore d = 2k, that is, d is even. 2.2 Asymmetric graphs Although we shall be mostly interested in graphs with nontrivial automorphism groups, let us briefly consider asymmetric graphs. Let P be a graph theoretic property such as ‘planar’ or ‘vertex-transitive’. Let rn denote the proportion of
- 46. 26 Various Types of Graph Symmetry labelled graphs on n vertices that have property P. If limn→∞ rn = 1, then we say that almost every (a.e.) graph has property P. We have already said that almost every graph is asymmetric. We shall soon prove a stronger result that will be used in a later chapter when we consider the Reconstruction Problem. The following probability space is often set up when studying random graphs. Let G(n, p) be the set of all labelled graphs on the set of vertices {1, 2, . . . , n} where, for each pair i, j, P(ij is an edge) = p and P(ij is not an edge) = 1 − p independently. Therefore a graph with m edges in G(n, p) has probability pm q(n 2)−m , where q = 1 − p. We shall need only this space when the probability p = 1 2 . In this case, each graph G in G(n, 1 2 ) has probability (1 2 )(n 2), which is, of course, equal to the probability of choosing G randomly from amongst all 2(n 2) labelled graphs on n vertices when all are equally likely to be chosen. Therefore, to show that a.e. graph has a particular property P one has to show that the probability that G ∈ G(n, 1 2 ) has property P tends to 1 as n tends to infinity. Now, let k be fixed. We say that a graph G has property Ak if all induced subgraphs of G on n − k vertices are mutually nonisomorphic. In other words, G has property Ak means that, if X, Y are two distinct k-subsets of V(G), then G − X G − Y. It is easy to show (Exercise 2.5) that if G has property Ak+1, then it also has property Ak and that if it has property A1, then it is asymmetric. We shall show that, for any fixed k, a.e. graph has property Ak. Lemma 2.5 Let W ⊆ V, |W| = t, |V| = n, and let ρ : W → V be an injective function that is not the identity. Let g = g(ρ) be the number of elements w ∈ W such that ρ(w) = w. Then there is a set Iρ of pairs of (distinct) elements of W, containing at least 2g(t − 2)/6 pairs, such that Iρ ∩ ρ(Iρ) = ∅. Proof Consider those pairs v, w ∈ W such that at least one is moved. (All pairs are taken to contain distinct elements.) There are g(t − g) + g 2 such pairs. For all but at most g/2 of these pairs, {v, w} = {ρ(v), ρ(w)} (the exceptions are when ρ(v) = w and ρ(w) = v). Let Eρ be the set of all such pairs. Then |Eρ| ≥ g(t − g) + g 2 − g/2 = g(t − g/2 − 1) ≥ g(t/2 − 1).
- 47. 2.2 Asymmetric graphs 27 Define a graph Hρ with vertex-set the pairs in Eρ and such that each pair {v, w} is adjacent to the pair {ρ(v), ρ(w)}. In Hρ, all degrees are at most 2. Degrees equal to 1 could arise because {ρ(v), ρ(w)} could contain an element not in W, and so the pair would not be in Eρ. Degrees equal to 2 could arise because {v, w} could be adjacent to both {ρ(v), ρ(w)} and {ρ−1(v), ρ−1(w)}. Therefore the components of Hρ are isolated vertices, paths or cycles. Let Iρ be a set of independent (that is, mutually not adjacent) vertices in Hρ. There- fore, for any pair {v, w} ∈ Iρ, {ρ(v), ρ(w)} is not in Iρ. Now, all isolated vertices in Hρ are independent, at least half of the vertices on a path are independent and at least one third of the vertices on a cycle are independent, the extreme case here being a triangle. Therefore |Iρ| ≥ |Eρ|/3 ≥ 2g(t − 2)/6, as required. Corollary 2.6 Let G ∈ G(n, 1 2 ), W ⊂ V = V(G) and |W| = t. Let ρ : W → V be an injective function that is not the identity. Let g = g(ρ) be the number of elements w ∈ W such that ρ(w) = w. Let Sρ be the event ‘ρ gives an isomorphism from G[W] to G[ρ(W)]’. Then P(Sρ) ≤ 1 2 2g(t−2)/6 . Proof Let Iρ be the set constructed in the previous lemma. Now, for a given pair {v, w} ∈ Iρ, the event ‘{v, w} and {ρ(v), ρ(w)} are both edges or nonedges’ has probability 1/2. These events, as they range over all pairs {v, w} ∈ Iρ, are mutually independent because they involve distinct pairs. But Sρ requires all these events simultaneously. Therefore, by independence, P(Sρ) ≤ 1 2 |Iρ| ≤ 1 2 2g(t−2)/6 , as required. The result of this corollary is the crux of the proof of the following theo- rem: There are too many independent correct ‘hits’ required for ρ to be an isomorphism, and the probability therefore becomes small as n increases.
- 48. 28 Various Types of Graph Symmetry Theorem 2.7 (Korshunov; Müller; Bollobás) Let k be a fixed nonnegative integer and let G ∈ G(n, 1 2 ). Let pn denote the probability that ∃W ⊆ V(G) = V = {1, 2, . . . , n}, with |W| = n − k and such that ∃ρ : W → V, ρ = id, ρ is an isomorphism from G[W] to G[ρ(W)]. Then, limn→∞ pn = 0. Hence, a.e. graph has property Ak. Proof Pick a particular W ⊂ V with |W| = n − k. This can be done in n n−k ways, and n n − k = n(n − 1) . . . (n − k + 1) k! nk . Let t = n − k. Let ρ : W → V be injective and not the identity, and let g = g(ρ) be the number of vertices of W that are moved by ρ. Let Sρ be the event defined in the previous corollary. Now, for a given value of g between 1 and t, how many functions ρ are there such that g(ρ) = g? Such a function is determined by the set {w : ρ(w) = w} and by the values it takes on this set. Therefore, there are less than n2g such ρ. Therefore, for a given fixed W, the probability of a nontrivial isomorphism is given by ρ=id P(Sρ) = t g=1 ρ:g(ρ)=g P(Sρ) ≤ t g=1 n2g 1 2 2g(t−2)/6 = t g=1 n2 2(2−t)/3 g t g=1 41/3 n2 2−t/3 g .
- 49. Discovering Diverse Content Through Random Scribd Documents
- 50. “Die olie word verkry uit die bessies van hierdie boom deur dit fyn te stamp en dan te kook; die bessie is net so groot as die kasterolieboom se pitte, dog is baie harder. Hierdie bessie braai Rondelyf uit saam met die melk van die gifbos. As dit afgekoel en koud is, dan is dit gereed om as olie vir verf te dien. Waar die soort gekookte olie op ’n klip val, maak dit ’n bruin vlek wat nie vandag, nòg môre, nòg oor jare daarna somar sal uitgaan nie. “My oorlede moeder het ons ou trektafeltjie met daardie soort verf geskilder. Op sagte voorwerpe, soos sagte hout, word die verf dof, maar op harde klip behou [145]dit sy helder kleur. Maar met al ons rondtrekkery het dit moeilik gegaan om ’n skaafplekkie aan die tafel te kry—dis ongetwyfeld, dié soort olie is goed. “Rondelyf het nog ’n ander soort verf gemaak, naamlik van ajoos, vet en harpuis. Die Boesmanlandse harpuis is baie skaars—so vat hy soms in plaas daarvan kors, of gom, van die t’kooibos. Met hierdie soort verf het hy soms sy hele liggaam besmeer—tot selfs sy bakkies. Dit het ’n mooi donkerbruin kleur. “Die ajoos is ’n soort plant nes die duiwelsbrood. [In Boesmanland word ’n paddastoel duiwelsbrood genoem.—Die Skrywer.] Maar waar die duiwelsbrood die vorm van ’n hoed of sambreel het, het die ajoos die vorm van ’n ronde bal. As dit ryp word, dan droog die vel uit, word hard; en as die vel dan oopgebreek word, dan is die hele bal van binne vol van ’n bruin, swart of geel poeier. Hiervan word die verf gemaak.” Tot so ver het oom Jan vertel. Die ajoos word in die Karoo en binnelande van die ou Kolonie oeltjie, of nambossie, genoem; en word sowel deur die Hottentotte as Boesmans gebruik om hulle wange mee bruin te skilder—soms word hulle gesigte nogal taamlik bont daarmee geverf. Ons het in ons kinderdae gesien dat blank meisies ook hulle gesigte tydelik daarmee geskilder het. Toe ons die rede daarvoor verneem, was die antwoord: “Dit hou die vel koel en as ons dit na ’n tyd afwas, dan is ons gelaatskleur weer lelieblank; want hier in die ope veld [146]brand die son ons heeltemal bruin.” Dit was natuurlik die geval waar die trekboere ver uit mekaar staan en besoekers nie aldag daar kom nie. Die meisies hou hul by sulke geleenthede eenkant.
- 51. VOORBEELDE VAN VERFKLEURE. Wel, om tot die verfkleure van die Boesmans terug te keer. Boesmans teken die Kaffers altyd swart af—daarvoor gebruik hul potswartsel; vir Boesmans gebruik hul meestal bruin—daarvoor gebruik hul bruin potklei, bruin klip wat ysterroes bevat; vir wit mense gebruik hul wit of liggeel klei, of skilferklip van dié kleur. Ook het ons skilderinge in groen en blou gesien, wat vermoedelik aan kopererts ontleen is. Soms vind ’n mens tekeninge wat amper rooi is; die verf hiervan word op seker plekke gegrawe. [149] [Inhoud]
- 52. No. 70.
- 53. Kinders van die Natuur. OPMERKINGS:—In hierdie Verhaal deel ons ’n paar gevalle mee hoe onverskillig ’n Boesmans die erns van die lewe kan opneem: plesier is sy grootste ideaal. Baie mense eet om te lewe; ander, weer, lewe om te eet en om die plesier van die wêreld tot hoofdoel te stel. Dit lyk of die Boesman hom beskou as net in die wêreld te gekom het om te eet en plesier te maak. Sy grootste genot is dan maar eet en dans. Hy kan nie lekker dans as hy honger het nie; en as hy sy bekoms aan ’n maal het, dan wil hy dans; en dit doen hy so goed dat almal wat hom sien dans, moet verwonderd staan. Dit het baiekeer gebeur dat as die baas van die plaas kuiergaste van ver kry, dat hy sy Boesmans roep om tot vermaak van die kuiergaste te kom dans—net soos die ou Farao’s van Egipte gedoen het om hulle dwergdansers tot vermaak van ander en hulleself te laat dans. Sê die oubaas: “Toe, ou Hans, kom dans vir die kuiermense die aasvoëldans,”— dan lyk ’n aasvoël ’n mooi bog om beter rond te spring en met sy vlerke rond te klap as ou Hans Boesman dit kan doen. Dan kom weer die bobiaandans, wanneer ou Hans Boesman se litte so los is as dié van ’n lamlendige houtpop. Daarna moet hy die [150]springbokdans uitvoer, en dan by hierdie geleentheid wip Hans soos ’n gomlastiekbal in die lug, skynbaar sonder ’n been, arm of spier te beweeg—want springbokke hou mos hulle bene en nek styf as hulle pronk. En as ’n klugspeler en nabootser het ’n Boesman nog nie sy gelyke gekry nie. Hy is ook glad nie punteneurig omtrent die soort kos wat hy kry nie. Dit kan maar goeie of slegte vleis wees, slange of skilpaaie, veldvrugte of uintjies—alles is welkom, as dit maar net nie doodmaak nie. Maar dan gebeur dit met groot droogtes dat kos skaars word en hulle hard moet stry om elkeen vir homself aan lewe te hou; dan maak hulle wonderlike planne om kos in die hande te kry. Loop hul dan kos op die lyf wat vir ’n week of meer kan dien, dan is daar geen voorsorg om matig aan te gaan en ’n stukkie vir die dag van môre opsy te sit nie. Nee, hulle eet gulsig tot hul nie meer kan nie, gaan slaap as dit dag is, of dans as dit nag is. Is alles op, dan bly die gedagte oor: “Ons het dit die dag of wat lekker en plesierig gehad.”
- 54. As honger en droogte diere teister, dan gooi hul hulle kleintjies weg; en wat doen die Boesmans? Mevr. Jacob de Clercq vertel aan ons dat toe sy nog ’n kind was, kom daar na haar vader, Gert van Niekerk, se plaas (waar nou Uitkyk-stasie, Middelburg, Tvl., is) ’n ou Boesman en sy meid; hulle plaas ’n klein meidjie wat nog nie kan loop nie, voor haar moeder en vader en soebat om tog maar die meidjie vir ’n jong os te ruil, daar hulle [151]baie honger ly. Ou mnr. Van Niekerk en sy vrou wou so ’n jong kind nie ontvang nie; dog die Boesmans bly aan soebat, hulle verlaat die werf stilletjies, gaan ’n bees vat en gaan daarmee voort. Mnr. Van Niekerk het die kind met ’n oorlamse Kaffer agternagestuur, maar hy kon die Boesmans nie vind nie—so het mevr. Van Niekerk maar die meidjie grootgemaak. Sy het haar ounooi getrou gedien en het tot haar oudag by die famielie Van Niekerk gewoon, en het naderhand sleg van gesig geword. Maar dit moet al baie droog wees as Boesmans nie meer iets te eet in die veld kry nie. Laat ons weer hoor wat oom Jan Visser ons vertel. Mnr. De Roubaix stuur die volgende mededeling aan ons:— Oom Jan Visser vertel: “Ek was eenmaal as kind met my vader mee in die Kalaharie-woestyn. Daar het ons eendag—kort voor sononder—’n klomp Boesmans teen ’n duin sien koes en opstaan. Hulle was druk besig om iets te soek en uit te grawe en steur hul nie aan ons nie. Rondelyf weet dadelik te vertel dat die Boesmans honger het en dat hul besig is om uintjies te grawe. Toe sien ons dat kort-kort een op sy hurke gaan sit en druk besig met grawe is. Dit het waarlik gelyk nes ’n klomp bobiane wat kos teen die rant soek. “Toe dit begin skemer word, het die Boesmans teen die duine gaan rus. Rondelyf vra toe somar verlof om daarheen te gaan, daar hy vermoed dat die hele nag fees sou gevier word. Hy was reg; want so was dit ook. Die Boesmans het hulle ramkies meegebring en het daarop die [152]hele nag gespeel. Die gedans was net lewendig en geesdriftig. Ons kon die musiek duidelik hoor, en daar is een riel wat hul toe gespeel het, wat ek self op Rondelyf se ramkie kon speel. Die naam van die riel is Die Bontperd. Dit het net twee of drie draaitjies en gaan min of meer so: Hor-tiek-tiek, hor-tek-tek, hor-tak!—dit word oor en oor herhaal. Toe Rondelyf terugkom, vertel hy dat van hierdie Boesmans het die poliesie van Upington en Kenhardt vroeër gaan haal en met die grootste gesoebat op
- 55. Upington gekry. Daar is hul toe gehuur geword om êrens anderkant Kenhardt aan ’n dam te gaan werk. Maar voor hulle die werkplek bereik het, het die laaste een weggeloop en die Kalaharie ingevlug. “Toe ek ’n kind was, het Rondelyf vir my meer as een ramkie gemaak. Hy het gewoonlik die hoepel van ’n vat geneem en rond gebuig dat dit so groot as ’n hoed se bol is. Aan die hoepel bind hy ’n stok wat as nek van die ramkie dien; bo-aan maak hy drie gaatjies waarin die pennetjies moet kom wat die snare moet aandraai. Hy neem dan ’n nat haarafgemaakte vel van ’n lam en trek dit oor die rondgebuigde hoepel en oor die stok so ver dit aan die hoepel vas is. Dan maak hy die drie snare aan die onderpunt van die stok vas, trek dié oor die romp van die ramkie en bind dit een vir een aan sy pennetjie vas. Dan maak hy ’n kam en plaas dit op die romp van die ramkie; dan lyk die affêre nes ’n drie- snarige ghitaar. Nou is ou Rondelyf my musiekonderwyser—ja, wat vang kinders nie aan nie! Ek het geleer om Die Bontperd [153]te speel, dan dans ou Rondelyf voor my en skoffel dat die stof so trek.” Tot so ver die verhaal van oom Jan. ’n Boesman sal hom die hele dag vreeslik vermoei om in die aand—al was dit maar vir ’n oomblikkie—’n plesier-plek by te kom. Ons weet van ’n geval waar ’n smous ’n mak springbok wat met die vee meeloop, geruil het. Die springbok kry dit in sy kop om terug te hol om sy pleegmoeder, die bokooi, te gaan opsoek. ’n Boesman bied sy dienste aan om die jong springbokkie vir ’n sopie brandewyn te gaan haal. Die smous beloof hom dit, en kort na sonop spring die Boesman weg en kom moeg die aand na sononder met die bokkie op sy skouer terug. Toe iemand hom vra of hy sy lyf dan oor ’n sopie so moet pla, antwoord hy: “Baas, ’n dag se swaarkry is niks, as ek net in die aand effe my lyf kan plesierig maak.” Die smous het egter ’n span roltabak by die sopie gevoeg; en meer voldaan en opgeruimd—toe die sopie begin te trek—het ons ’n skepsel nie gesien nie. Met so ’n beloning sal hy nooit staak nie. Oom Piet Smit het ’n Boesman en sy seun gehad wat snags na die skape in die bosveld moes kyk. Hulle weet as hul durf die veewerf in die nag verlaat, dan kry hul die volgende dag ’n afgedankste afranseling. Maar daarom gee hul nie as hul net in die nag die bierpotte van die Kaffers wat op ’n afstand woon, kan bykom. Hul steel enigiets om vir bier te gaan verruil. [154]
- 56. Jagters in die Kalaharie kry baie dienste uit die swerwende Boesmans net deur hul met ’n stukkie tabak te vergoed. Doen hul dit nie, dan moet hul van dors omkom. Dis wonderlik dat die wildste Boesman van tabak gehoor het en baie uit sy pad sal gaan om dit in hande te kry. [157] [Inhoud]
- 57. No. 71.
- 58. Fyn Diefstalle. OPMERKINGS:—Nou sal ons enige staaltjies gee hoe fyn ’n Boesman kan steel sonder uitgevind te word—maar baiekeer loop hy hom tog vas, daar die witmense te goed met sy planne bekend is. Daar is baie mense wat volhou en nie van die punt af te kry is nie, nl. dat ’n Boesman ’n gebore dief is. En tog sit in so ’n algemene verklaring nie altyd waarheid nie. Hoeveel mense vertrou hulle goed aan die sorg van Boesmans toe en vind hulle vertroue nie misplaas nie? Ons was in ’n huis waar die Boesmanmeid aan die nooi sê: “Nooi, ek sal my hande nie aan jou goed sit nie; maar ek beloof nie om nie te steel as ek by jou drank uitkom nie.” En die huisvrou het ons verseker dat nie ’n speld verlore geraak het nie, maar die drank kon sy nooit te veilig onder slot gehou het nie. Andermaal gaan ons vertel wat oom Jan Visser verhaal het en aan ons deur mnr. E. de Roubaix toegestuur is:— “Eenmaal het ’n veekoper met sy vee by ons staanplek gelê. Die vee kom gewoonlik oor ’n kaal vlakte met klein bossies begroei na hulle lêplek toe. As ’n haas oor die veld loop, kan ’n mens hom sien. Wel, soos gewoonlik kom die [158]vee weer een agtermiddag oor dieselfde vlakte wei-wei veewerf-toe; en die veekoper se twee wagters was by. “Die volgende oggend, na ou gewoonte, word die skape getel, en toe word bevind dat een aan die getal kort kom. “My oorlede vader merk toe teen die veekoper op: ‘Dis baie snaaks dat net een skaap aan die getal ontbreek—een skaap sal mos nie afdwaal nie! Ons het seker sleg getel, of anders is die skaap gesteel of aan die geilsiekte dood.’ “Hulle tel weer die skape; maar nog word een vermis. Toe besluit hul om met die honde effe in die veld rond te soek. Hulle was nog geen paar honderd tree van die tent af nie of hul was op daardie kort blomkool-vlakte. Hulle sien dat die honde so gretig in ’n droë bos snuffel. Na nader te gestap het, bevind hul dat dit ’n droë kraalbos is wat iemand in ’n erdvarkgat se bek geprop het. Hulle trek die bos uit, loer in die gat—en daar vind hul die skaap afgeslag, en die vleis,
- 59. behalwe die stert en rugstring, was in die vel toegedraai en aldus in die gat sorgvuldig versteek. “‘Dis niemand anders se werk hierdie nie as dié van Rondelyf,’ roep my vader uit. ‘Ek sal die hele ding gou-gou vir jou uitvind,’ sê hy toe hulle huis-toe stap. “Toe hul daar aankom, pak hul Rondelyf aan sy arm en bind sy hande agter sy rug vas en lei hom na die klein handskroefie om sy oor daarin vas te draai. Want ’n Boesman moet ’n mens gou op die lyf loop om hom bang te maak en nie kans gee om te dink nie. “‘Kom, Rondelyf, vertel die hele waarheid; hoe het jy die skaap daar op die kaal vlakte gevang sonder dat ’n [159]mens of die twee wagters jou gesien het?’ vra my vader toe hy maak asof hy die skroef met geweld gaan aandraai. ‘Toe, kom uit met die waarheid, of ek draai jou ore en vingers in die skroef af!’ “Toe kom hy sonder te versuim met die hele waarheid uit, en vertel dat hy in die agtermiddag al in die erdvarkgat met ’n bossie op sy kop gebind, gaan lê het; want hy weet dis die huistoekom-pad van die skape. Toe die skape om die erdvarkgat wei, het hul hom nie daar binne-in gewaar nie. Hy het sy vet hamel uitgesoek, wat geen tree van die gat af was nie, het sy hand uitgesteek, hom gegryp en in die gat gesleep. Die skape om die gat het wel ’n bietjie geskrik; maar dit doen hul heeldag as hul op ’n hasie, slang of skilpad afkom. Dadelik was alles oor; die skape trek die gat verby sonder dat die twee wagters geweet het dat daar iemand en ’n skaap binne in die gat gewees het. “Na hierdie verklaring was Rondelyf gou uit die skroef verlos; dog my vader het hom ’n afgedankste loesing afgetel omdat hy durf die goed van ’n ander steel. Dit het vir Rondelyf en sy maats skaars ’n week gehelp, dan het hul die taaiste pak slae vergeet en steel weer van vooraf.” Mnr. Piet Venter het ons vertel dat sy vader, toe hy nog in die binnelande van die ou Kolonie gewoon het, twee getroue honde gehad het. Hulle het baie jakkalse en ongediertes doodgebyt; en ’n Boesman het nie gedurf om naby te kom nie. Hulle het ook ’n getroue ou Boesman, ou Ertman, gehad, wat na die koeie met jong kalwers kyk. [160] Eendag slag my vader, mnr. Piet Venter, ’n vet koei; die vleis moet in die nag in ’n boom hang, en die twee honde pas dit op. Dog die ander dag makeer ’n
- 60. voorkwart; en hul kon nie uitvind hoe dit onder die oë van die honde uit gesteel is nie. Later, op ’n wonderlike manier, vind hul uit. Ou Ertman het aan sy maats in die veld vertel dat sy baas ’n koei gaan slag. In die nag het een in ’n boom ver van die huis geklim en nes ’n jakkals geskree; solank die een trag die honde na hom, die sogenaamde jakkals, te lok, het die ander êrens naby verskuil gelê. Die honde hol na die sogenaamde jakkals toe en blaf daar; onderwyl steel die ander Boesmans die vleis en hol daarmee veld-in en vlug ver die berge in; dog hulle het nie vergeet om stilletjies in die weiveld aan Ertman sy beskeie deel van die vet vleis te bring nie. [163] [Inhoud]
- 61. No. 72.
- 62. As nood die Boesmans druk. OPMERKINGS:—Hier gee ons ’n paar voorbeelde hoe ver nood iemand kan drywe. Die boesman, as kind van die woestyn, het ’n harde stryd teen wilde diere, vyande, honger en dors. Selfverdediging is die eerste wet van bestaan. En daar is so baie maniere van selfverdediging. Ons het al vertel hoe Boesmans soms hulle kinders vir kos verruil. Dit is nie alleen om hulle lewe te red nie, maar ook dié van hulle kinders. Maar hoe maak hul met hulle swak ou volk wat nie meer met jag kan meedoen nie as die nood hul dryf om te trek—ja, na ’n veraf geleë streek te trek? Hieromtrent het ons informasie ingewin van mnr. W. A. van Zyl, Klein-Breipaal, Boesmanland. Dit deel hy ons mee: “Die wilde Boesmans het drie maniere om van hulle ou volk ontslae te raak as die nood hul erg druk, sodat hul verplig word om die streek te verlaat. “Die eerste manier is om ’n kliphok om die swakke en oue van dae te pak. Daarin plaas hul dan kos en water om vir ’n geruime tyd te dien. Dan trek hul weg en laat die ou persoon verder aan sy eie lot oor. Die kliphok, wat hoog en sterk gepak is, is natuurlik bedoel om die ou skepsel [164]teen verskeur van wilde gedierte te beskerm. As hy binne-in veilig is, dan word hy van die vrees en angste gespaar om deur wilde diere wreed gedood en opgevreet te word. “Die twede manier is: Hulle grawe ’n diep ronde gat regaf—natuurlik waar geen klippe te vinde is nie—, plaas die ou daarin, en ook kos en water, maar dan vir ’n kort tyd. As hul dan nie weer terugkom voor hy dood is nie, dan smeer die meide hulle hare vol verf met vet gemeng. Hulle neem dan klein klippies en gooi dié rondom sy lyf in die gat totdat hy hom nie meer kan roer nie en daar vassit; dan verlaat hul die plek, om nie na die graf terug te kom nie. Die ou kan hom dus nie roer nie en moet so van ellende sterwe. “Die derde manier is: As hul op trek is en een van die oues beswyk van ouderdom of dors, dan maak hul ’n langwerpige hol plek in die grond, lê gras daarin en plaas die ou daarop: daar moet hy lê tot hy dood is—nou is daar geen kos of water om aan hom agter te laat nie.”
- 63. Tot so ver het mnr. Van Zyl ons vertel. In ieder geval waar hul hulle oues agterlaat, is ons deur Boesmans verseker geword dat hul altyd ’n kliphok, waar klippe te vinde is, en ’n doring-heining, waar doring-bosse te vinde is, pak, om die agtergeblewene teen verskeuring deur roofdiere te beskerm. Dit alles lyk wreed; maar ’n Boesman het ons die vraag self gestel: “Sal baas die hart hê om jou eie pa of ma [165]koelbloedig te vermoor as daar net liefde en nie rusie is nie? Wat help dit om agter by hulle te bly? Dan sterf ons met vrou en kinders almal tesame—en wat word dan van die Boesman-nasie?” Omtrent wat rein en onrein is, bestaan by volkere groot verskil. Ons eet b.v. skaap- of beesboutvleis, snoek, hase, vark, ens., terwyl ’n Jood dit as onrein ag. Ons eet mossels, klipkous, maar nie seekat nie, terwyl die suidelike nasies van Europa seekat eet omdat dit aan die skulpsoort behoort. Ons eet ystervark, wat ’n knaagdier is, maar verfoei muise, wat ook knaagdiere is. Ons eet nie perde en haaie nie, terwyl dit in Europa wel geëet word. Ons eet nie kruipende diere nie, maar wel skilpaaie. Ons eet nie paddas nie, en in Frankryk is dit ’n lekkerny. En so kan ons nog baie goed opnoem, b.v. dat Kaffers vis as onrein beskou, maar hulle sal ’n jakkals eet. Baie mense wil nie wild eet nie of dit moet ’n luggie hê. Nou, die Boesman steur hom aan niemand nie—hy eet wat voorkom, soos vleis, vis, muise, paddas, slange, akkedisse, skilpaaie, wurms en vleis, of dit vars of oud is, miere en alles—as dit net nie doodmaak nie. Hy gaan dus met almal saam. Want die honger en dors van die woestyn het hom daartoe gedwing—net soos ons in die geskiedenis lees dat as ’n bemuurde stad beleg is om die bevolking uit te honger, dan het die mense net alles geëet en gedrink wat ’n Boesman eet en drink. Maar as hy goeie kos kry, bepaal hy hom meer daarby as tot die ander, [166]en beskou dan b.v. slakke en paddas in dieselfde lig as die Franse. Deurdat ’n stomp pyl—soos dié van ’n Boesman—nie deur die dik vel van groot wil kan boor nie, is die Boesman verplig om met gifpyle te skiet. Maar hoeveel lewens van hulle het dit gekos voor hul die kuns goed verstaan het om te weet watter gif te gebruik en hoe om dit te gebruik? Die nood het hom dus tot gif gedrywe. [169]
- 64. [Inhoud]
- 65. No. 73.
- 66. Wat ’n Boesman alles kan doen. OPMERKINGS:—Ons noem hier enige feite wat ’n ander nasie moeilik ’n Boesman sal nadoen. Vra aan iemand wat so te sê aldag met Boesmans omgaan, of hy ooit ’n Boesman ontmoet het wat verdwaal het; en hy sal ronduit verklaar dat hy van so ’n geval nie weet nie. As ’n Boesman eenmaal op ’n plek gewees het en weet waar dié plek geleë is, dan sal hy van enig ander oord reguit met ’n onbekende pad na die plek toe loop—al was dit in die donker. In hierdie opsig is hy net soos ’n posduif wat oor honderde myle deur mistige weer huis-toe vlie. So vertel mnr. H. Visagie ons dat toe hy ’n jongeling was, het by sy vader ’n ou Boesman en sy gesin gewoon. Die ou het maar lus gehad om noordwaarts na die Kalaharie te trek. Na hy vir maande en maande weg was, kom op ’n goeie dag een van die ou se kinders aangestap—hy het na sy oubaas toe weggeloop. Volgens die twaalf-jarige Boesmantjie beduie, het hy met sy ouers eers noord- wes getrek, toe weer reg sonop; en nadat hy maande met hulle rondgeswerf het, het hy weggeloop en reguit deur die veld koers na sy oubaas gevat. Hy was vir dae op pad, het baie honger en dors gely, en is deur ’n streek gegaan waar [170]leeus en wolwe ronddool. Toe hom gevra was hoe hy die pad reggekry het, was sy enigste antwoord: “Ek weet mos waar oubaas woon.” Wel, sou ’n kind van twaalf jaar van ’n ander nasie dit reggekry het om die so onbekende weg te vind, om honger en dors so te trotseer en om die vermoeienis deur te staan, sonder eens van die gevaar van ongediertes te praat? Ons het self gesien hoe hul die warmste van die dag uitkies om wild te agtervolg —wat beteken dat hul ’n harder natuur besit as ’n dier. So skryf mnr. E. de Roubaix ons verder wat hy uit die mond van oom Jan Visser aangeteken het:— “My vader het vir Rondelyf meestal gebruik om verlore vee op te spoor—en hiermee het hy sy weerga nie kon vind nie. Hy kon spoorsny beter as ’n gewone hond en kon beter sien as ’n aasvoël uit die lug; want die geringste verskuiwinkie ontsnap sy blink ogies nie. Daarby het hy ’n groot stuk wêreld in ’n omsientjies kon afdraf sonder enige tekens van vermoeienis te toon. Meer as een middag, as dit regtig warm is, het hy wilde volstruise van agteraf ingehardloop en gevang. By hom het ek as kind dikwels volstruisbiltong geëet.
- 67. “Hy was ’n opregte Boesman; maagseer, koppyn, jig, of tandpyn het hy in sy hele lewe nie gekry nie. En soos hy is, so is al die Boesmans.” Ofskoon die Boesman se gesig goed is, is sy ruik enigsins in seker opsig gebrekkig; daarom kan hy die stank van aas of sleggeworde vleis goed verdra. Selfs die [171]spelonk waarin hy woon, het ’n walglike walm wat enig witmens sou terugdrywe. En wat nog meer is, is dat hy in so ’n omgewing nie die pes kry nie; sou witmense onder sulke omstandighede die lewe moet voortsit, dan lê almal binne ’n week of so plat aan die koors. Wat hy nog meer kan doen, is om kaal op nat grond te slaap. Soms is sy skuilplekkie in ’n renerige nag of dag net om te troos, maar nie om die instromende water uit te hou nie—en tog weet hy van geen jig of inflammasie nie. Ons het reeds vermeld watter fyn planne hy met jag kan maak. Mnr. W. A. van Zyl, van Klein-Breipaal, deel aan ons mee:— “Boesmans bekruip wild seer fyn. Solank die wild nog op ’n afstand is, kruip hy hande-vier-voet; dog kom hy nader, dan seil hy op sy buik. Al wat hy moet doen, is om ’n bossie op sy kop vas te maak om sy swart hare te bedek; want dit is al wat die wild kan beken, daar sy lyf dieselfde kleur as die grond het. Hy is dan in staat om so na aan die bok te kom as hy verkies—net, hy moet onderkant die windkoers bly, sodat die bok die bekruiper se ruik nie kry nie. Dieselfde plan volg hy as hy wild voorlê. Hy kruip dan gewoonlik tot op ’n afstand tussen vyftien en vyf-en-twintig tree, hy kan deur sy fynheid nog nader kom, maar die pyl moet ’n seker distansie trek om sy snelheid te kry.” Iets wat ’n Boesman ook beter as ’n ander kan doen, is om weg te kruip. Eens het ons ’n plaas gemeet waar twee Boesmantjies kalwerwagters was. In ’n grasvlei het ons ’n vlag gesteek en het die Boesmantjies belet om met die [172]kalwers naby die vlag te wei. Hulle het hul egter daaraan min gesteur, en die vlag is verskeie male deur die kalwers uit die grond gepluk geword. Die baas se twee seuns en ons persoonlik sien die Boesmans in die grasvlei rondspeel; ons al drie stap daarheen—maar daardie Boesmantjies kry, was net so goed om kwarteltjies daar te soek, en tog, die gras was nie hoër as twee voet nie. Ons het vir die pret lank gesoek, maar moes so onverrigtersake omdraai. Toe ons omtrent aghonderd tree weg was, sien ons hul daar weer rondspeel.
- 68. Hierdie eienskap help die Boesman om as jagter uit te blink, en om in ’n minderheid teen ’n oormag te veg. Hierdeur was hy in staat om sy grond vir so ’n lang tyd teen Hottentot, Kaffer en witmense te verdedig. Eenmaal het ’n ou Kaffer op die westelike grense van Transvaal hom aldus op sy eienaardige manier aan ons uitgedruk:— “My baas, die Boesman is somar die ding. Jy sien, baas, jy en ek maak die baklei: jy kyke my oog, ek kyke jou oog. Maar die Boesman!—nee, my baas. Jy pla net verniet jou oog om vir hom te soek. Soos jy soeke die wind, jy kry vir hom; soos jy soeke die bobbejaan, jy kry vir hom; maar die Boesman! hoe jy sal weet hy is daar? Ja, al wat jy sal wete is: hy is daar, en hy stuur die dood na jou toe.” Die Boesman het die jagveld en die waters as sy eiendom beskou. Die wild is sy vee en dit word hom afgeneem; so moor hy ook onder die vee van ander. Hy weet vir hom is daar geen genade nie; waarom moet hy [173]die lewe van ander in ’n geveg spaar? Hy of sy vyand moet dood. As daar maar ’n dosyn vegmanne in ’n bergstreek woon, dan is dit nie elkeen se werk om hulle daaruit te verdrywe nie; en wie sal hom in die woestyn gaan agtervolg? Sy geheue is baie goed om plekke te onthou waar hy die doppe van volstruiseiers vol water begrawe het. Dit bly daar lank verberg; want mnr. W. A. van Zyl skryf aan ons:— “Hulle woon gewoonlik ver van water. Die water word in volstruiseierdoppe aangedra na hulle werf. Ook begrawe hul orals op sagte plekke sulke doppe vol water om in tyd as hul vlug of trek te gebruik. Ek het self in my kinderdae op sulke gevulde doppe afgekom as die wind hul in die duine oopwaai.” Ja, en ons wonder hoe water wat solank in ’n eierdop gestaan het, moet smaak! Ons sal in die volgende boekdeel ander ware gebeurtenisse meedeel wat haas ongelooflik is, en sal aantoon wat ’n Boesman nog meer kan doen. [177] [Inhoud]
- 69. No. 74.
- 70. Wat ’n Boesman nie kan doen nie. OPMERKINGS:—Uit hierdie Verhaal blyk dit dat die Boesman steeds ’n raadsel vir die ontwikkelde mens bly; hoe meer ons hom bestudeer, hoe raadselagtiger is hy. As nasie van jagters wat net leef van wat hul gedurende die dag versamel het en wat hul meen die volgende dag weer in te samel, en ook dat hul geen vee van hulle eie aanhou nie, so het die Boesman hom nie erg bekommer oor getalle nie. En daar hy van getalle min verstand het, kan hy ook nie syfer of reken nie. En tog, daar sal geen pyl van hom wegraak nie, of hy weet dit. As hy bokwagter is, sal daar geen bok vermis raak nie, of hy weet dit. Al lyk lammertjies hoe eenders, tog weet hy om aan iedere ooi haar regte lam te besorg. Van die geldwaarde van iets het hy hoegenaamd geen begrip nie; en as ’n mens hom geldstukke wys en die name en waarde daarvan opnoem, sal hy dit wel leer; maar meng ’n spul silwergeld, soos trippense, sikspense, sielings, tweesielingstukke en halfkrone deurmekaar en vra hom hoeveel dit alles tesame uitmaak, dan is hy nie in staat om die waarde daarvan op te gee nie. Daarom word Boesmans vir kos, tabak en ou klere gehuur—hulle verkies dit bo geld, waarvan hul niks verstaan nie. [178] Ek het eenmaal ’n Boesman gehuur wat ’n hele ruk op Kimberley gewerk het. Hy kon daar met die wette maar nie regkom nie, en was daar tweemaal in die tronk; so is hy daar weg. Ek sê aan hom dat ek hom kos en tabak sal gee en vyf halfkrone in die maand—ek wys hom op my vingers hoeveel vyf is. Hy skud sy kop en sê: “Nee, baas, ek het nog altyd op Kim’ley meer verdien; die base daar het elke maand by elke vinger ’n sieling neergesit; en as baas dit ook wil gee, sal ek by baas mooi werk; so nie, dan gaan ek ’n ander baas soek.” So het ek hom vir tien sielings in plaas van twaalf sielings en sikspens per maand gehuur. En hy het my baie trou gedien; maar dat ek aan sy verstand kon bring dat vyf halfkrone meer as tien sielings is, was net verniet. Wie die Goewerment is, kon hy maar nie verstaan nie. Hoe ek hom ook al die vorm van goewerment uitlê, hy kon daar geen begrip van kry nie—hy meen dit moet ’n ryk man wees wat alles besit en oor algar baas is. Hy vertel toe:— “Met ons Boesmans daar in die veld is dit só gesteld: Niemand is oor ons baas nie. Net ons pa en ma as ons nog kinders is; maar hul sal vergeet om aan ons
- 71. te vat as ons self ’n mannetjie is wat met ’n kierie ’n breekskoot kan slaan. As iemand oor ons wil kaptein speel, nes by die Hottentots en Kaffers, pak ons vir ons eenvoudig in en laat hom waar hy is. Hy sal dit vir eenkeer en tweekeer probeer, maar vir die derde keer sal hy gou leer dat hy nie met kinders te doen het nie. Is dit dan nie reg nie, baas? Kyk, al die wild is mos ons algar syne saam, die veldkos [179]is ons syne saam, die water is ons syne saam; en wie het hom daaroor baas gemaak? Al wat ons ons eie noem, is ons vrou en kinders, ons hond en gereedskap—daar moet ’n ander die pote van afhou: dis ons eiendom—nie waar nie baas? Dis mos reg so, nè baas?” Hieruit is dit duidelik dat ’n Boesman nie kan regeer nie en is self moeilik om te regeer—mits ’n mens sy maniere goed bestudeer en hom daarna behandel; dan is hy getrou aan sy baas en nooi. Om ’n Boesman te leer lees en skrywe, is net tydverkwisting. Toe ons persoonlik veewagtertjie was, het ons smiddags onder die bome in die somer baie tyd gehad om na Hans Boesman se stories en vertellings te luister. Ons het letters in die sand gemaak om hom ’n begrip te gee hoe die witmense lees en skrywe. Hy wou ook probeer om die letters te leer, en het nogal behae daarin geskep; maar hy moes dit opgee: “dis tog te swaar om te leer,” het hy gesê, en so was die skoolganery op ’n end; terwyl ons Basoetoe-skaapwagter binne ’n paar maande uit sy boek kon lees. Dit het ons van persoonlike ondervinding; want ons kinders het vir Hans as ’n interesante speelmaat beskou en wou hom aanhelp. In die jaar 1799 het eerw. J. J. Kicherer van die Londense Sendinggenootskap aan Sakrivier, Boesmanland, ’n sendingstasie aangelê, met die spesiale doel om die geestelike belange van die Boesmans te bevorder. Aan hulp het dit nie ontbreek nie. Die boere het beeste, skape en tabak aan die sendeling gegee om daarmee die Boesmans na Blyde [180]Vooruitzigt—soos die naam van die sendingstasie was—te lok. Die Boesmans het by hope gekom en daar gebly tot daar nie meer beeste, skape en tabak gekom het nie; toe het hul weer na die wilde lewe die toevlug geneem; want werk—soos Hottentots en Kaffers by sendingstasies—wou hul nie. Nou, dis gewoonlik dié Boesmans wat ’n tyd onder die witmense gewoon het en wat weer die wilde natuur aanneem, wat die groot kwaaddoeners is. Mnr. W. A. van Zyl, Klein-Breipaal, noem hul wildehonde. Dit skryf hy aan ons omtrent die verwoesting deur hulle aangerig:—
- 72. “Die Boesman se geaardheid is nes dié van ’n wildehond. As hulle vee gesteel het, dryf hul dit op ’n draf vinnig weg tot by ’n ruie plek of klipkoppe, en dan maak hul almal, grootvee sowel as klein vee, voor die voet op ’n hoop dood. So het hul troppe en troppe vee van ons doodgemaak—so maak wildehonde ook. Die wilde Boesman sorg nooit vir die dag van môre nie: wat hy nie kan opeet nie, kan maar bederwe en vergaan; môre doen hy weer dieselfde.” Dit het wel gebeur as vee te moeg word en nie vinnig genoeg na die sin van die Boesmans voortgaan nie, dat hul dan sulke diere die haksenings afsny of hul doodmaak; maar gesond wil hul die dier nie laat agterbly nie. Dit het ook gebeur dat hul van die dooie vee vleisskanse maak om agter te lê en hulle so te verdedig. Wanneer daar so ’n oorvloed van kos is, dan gun hul die leeu ook daarvan, wat ook vleis vir hulle uit sy oordaad laat bly—die wolwe [181]en jakkalse kan ook maar snags kom bysit. Wat dan nog oorskiet, is die volgende dag die Boesman syne—hy is nie inhalig nie, en gun aan die ongedierte ook iets. Wel, as ons al sulke dinge in oëskou neem, dan laat dit ons ver dink; want ons sien dat ons hier met die voorgeskiedenis-mensdom te doen het. Dit het duisende jare geneem om die barbare uit die mensdom te beskawe. En hoe sal ons dit regkry om meteens van ’n lae, gesonke mensheid in een geslag ’n beskaafde mens te maak? Beskawing klim van een geslag op na die volgende. En tog, met dit al is die Boesman op seker punte van vooruitgang vir die Kaffers en Hottentots voor, soos bv. hulle begrawe altyd hulle dooies (met gesig sonop), terwyl Hottentots hulle dooies in erdvarkgate stop, en sommige Kaffers die lyke êrens plaas sonder om dié te bedek. In tekenkuns het hul ook verder gevorder. [185] [Inhoud]
- 73. No. 75.
- 74. Wat alles in voordeel van Boesmans is. OPMERKINGS:—Ons gee nou seker punte aan wat steeds gedien het om die bestaan van Boesmans vir soveel eeue—nee, duisende jare—te waarborg. Daar bestaan geen nasie op aarde nie, of hulle voorouers was vir duisende jare terug barbare wat sonder klere rondgeloop het, wat geen ander werktuie kon maak nie as van klip, hout en horings; hoe om metale te bewerk, het hul niks van afgeweet nie. Sommige nasies het eerder as andere tot beskawing gekom, soos die ou Egiptenare, Babiloniërs, Feniesiërs, Grieke en Sinese; dog die Engelse, Franse, Duitsers, Hollanders, Russe en Skandinawiërs het later gevolg, en het toe voordeel van die Griekse en Romeinse beskawing getrek. Wat ook al mag gegis word deur geleerdes omtrent die oorspronklike geboorteland van die Boesman, dit weet ons as ’n onbetwiste feit dat Afrika hulle land was en is. En Afrika, met sy groot inlandse woestyn, het die noordelike beskaafde volke teruggehou; so het die Boesman die voordeel gehad dat hy sy rol eenkant kan vry uitspeel. ’n Ander voordeel vir hom is dat Afrika die rykste wêrelddeel vir wild is. Hy het dus nie nodig gehad om met mak diere opgeskeep te wees nie. Ook die veld het genoeg wilde vrugte en veldkos vir ’n jagter opgelewer, [186]sodat hy nie nodig gehad het om self te plant en te saai nie. Het hy lus vir vis, dan maak hy eenvoudig fuike van latte en lê dié in die riviere, en hy kry oorgenoeg. Op hierdie manier het hy verkies om in ’n lui-lekker-land te woon. Daar is niks verédelends in jag nie—dit beteken die vermoor van onskuldige lewens; en om ’n naam te maak, word soms meer wild gedood as wat vir die behoefte nodig is. So het die Boesman hom nie uit sy barbaarsheid verhef nie en het stil bly staan sonder verstandelik te ontwikkel; want ’n redelose dier, soos ’n leeu, kan ook jag en genoeg vind om van te lewe. Dan is dit ’n natuurwet wat die Skepper daar gestel het, dat die geskikste altyd sy pad en bestaan beter vind as ’n minder geskikte. Die geskikste ontwikkel kragtiger en laat meer nakomelinge na as die minder geskikte, wat nie behoorlik vir sy nakomelinge kan sorg nie. Dit op die Boesman toegepas, is dit duidelik dat die vader wat die beste kan hardloop, beste kan sien en die gepaste kleur van vel het om hom net soos die veld te laat lyk—dié vader bring ’n kragtiger
- 75. famielie op; terwyl die swakker en siekliker man miskien nie eens ’n vrou kan kry nie; en kry hy een, dan is sy swakker kinders nie bestand teen die harde eise van die woestyn nie: hulle sterf en laat ook nie ’n swak geslag na nie. So het net bly voortlewe wat vir die lewensstryd geskik is. Laat ons die Boesman nou beskou soos hy vandag is, en let op wat tot sy voordeel is. [187] Sy kleur is net soos die veld waarop hy jag. Mense het reeds daarop gelet dat daar ’n klein verskil in kleur is tussen die Boesmans wat in die Kalaharie woon en die Boesmans wat suid van Grootrivier gebore en opgegroei is. Daarom het mnr. W. A. van Zyl in die vorige verhaal gesê dat al wat ’n Boesman nodig het om te vermom, is sy swart hare, wat deur die wild kan beken word; sy kleur is net soos dié van die grond. Aldag is jagdag, maar nie vangdag nie. So gee dit eienskap dat daar dae van oorvloed is, en dae van gebrek, wanneer honger en dors die jagter en sy famielie kwel. Hieraan het die Boesman se maag gewend geraak en sy liggaam het in dié rigting ontwikkel om in tyd van oorvloed voedingskrag op ’n seker deel op te gaar. Wie ’n Boesman ken, weet dat ’n seker deel van sy lyf besonder fris ontwikkel is. Dus moet ons die tekenaar nie blameer nie as hy ’n Boesman net so afteken as hy in die natuur is. Mense wat in stede en dorpe van kindsdae af woon, kan gewoonlik nie so ver sien nie as mense wat op die platteland woon, omdat hul meestal in strate loop wat met geboue begrens is; dus het hul geen ver uitsig nie, en die oog word geoefen om naby te kyk. Die Boesman, as jagter, moet ver en naby oor die vlaktes en teen rantjies sy oë laat dwaal; dus is hulle oë van geslagte tot geslagte geoefen om ver en naby te kyk. Dis daarom dat sy gesig so goed as ’n vèrkyker is. Hy het dus die voordeel van ’n vèrsiende oog. Sy ooglede vorm twee skrefies, omdat hy oor vlaktes in skerp lig kyk. [188] Iemand groot en swaar van gewig kan nie so vinnig hardloop en is ook nie so rats nie. Slaan maar ag op wat ’n kind in ’n dag afmaak, probeer dit alles na te doen, en voel of jy in die aand nie doodmoeg gaan slaap nie. Vergelyk jou krag met dié van ’n kind, en staan dan verbaas. Net so met die Boesman: deur hy klein is, weeg hy nie swaar nie; die spiere van sy bene is goed ontwikkel. Die beste hardlopers onder hulle is ook die beste jagters—dus: die beste versorgers
- 76. van hulle famielies; en so is hierdie eienskap van geslag tot geslag onder hulle voortgeplant, net soos dit die geval is met windhonde en resiesperde. Die hardste en vertroubaarste werkers en arbeiders van die wêreld is persone met ’n vaste liggaamsbou en ’n gemiddelde grootte van ongeveer ses-voet- hoogte. Op hierdie eienskap kan die Boesman nie aanspraak maak nie. Dit is dus wreed om van hom swaar liggaamsarbeid te eis. Maar gee hom vee om mee te werk, of werk van ’n minder vermoeiende aard, dan is hy daar tuis net soos enige ander. Wat die Boesman deur al die eeue van uitroei of uitsterwe bewaar het, is dat daar selde of nooit ’n getroude meid gevind word wat nie ’n aantal kinders het nie. Ons oumense het reeds in vroeër dae opgemerk dat die Boesmans so snel vermeerder. Maar nou in ons dae is hul aan die verminder. Die rede hiervoor is dat die Hottentots en Kaffers, wat van die noorde die land oorstroom het, die mans vermoor en die meide vir hulle tot vroue geneem het. En in ons dae wil ’n Boesmanmeid liewers ’n Hottentot of [189]Kaffer tot man hê. Dit is dus die meide wat geen nasionale gevoel vir hulle eie nasie besit nie; sy is die groot oorsaak van dit alles; haar verval is die ondergang van haar volk. Haar volk is in die minderheid en word deur die ander verag, waaroor sy ’n seker mate van skaamte gevoel; en dit maak dit vir haar so gemaklik om haar eie nasie te verstoot en haar met ’n vreemde nasie te vereenselwig. Maar sou die Boesmanmeid soveel respek hê om trou by haar volk te staan, saam met hulle die lief en leed te deel. dan sal sy die redmiddel wees om die bestaan van die Boesman voort te plant. Maar noudat die Boesmanjong nie trou nie, sal dit nie meer baie geslagte duur nie of die Boesman sal tot die verlede behoort. [190]
- 77. Kunstenaar op ’n Studiereis na die Boesmans in die Kalahariewoestyn.
- 78. Kolofon Beskikbaarheid Hierdie e-boek kan deur enigiemand gebruik word, oral, gratis en met weinige beperkings. U mag dit kopieer, weggee of hergebruik in terme van die Project Gutenberg Lisensie hierby ingesluit, of aanlyn by www.gutenberg.org . Hierdie e-boek is gelewer deur die aanlyn Distributed Proofreading span by www.pgdp.net . Metadata Titel: Boesman-Stories: Deel III. Die Boesman self, sy sedes, gewoontes en bekwaamhede Skrywer: Gideon Retief von Wielligh (1859– 1932) Info Illustrator: Hans Anton Aschenborn (1888– 1931) Info Taal: Afrikaans Oorspronklike publikasiedatum: 1921 Sleutelwoorde: San (African people) -- Folklore Tales -- Africa, Southern Hersieningsgeskiedenis 2020-09-26 Begin. Eksterne verwysings
- 79. Hierdie Project Gutenberg-e-boek bevat eksterne verwysings. Hierdie skakels werk moontlik nie vir u nie. Regstellings Die volgende regstellings is aan die teks gemaak: Bladsy Bron Regstelling Redigeerafstand 11 afbeel afbeeld 1 84 skerppuntbeenbyle skerppuntbeenpyle 1 103 Geskeidenis Geskiedenis 2 113 sy my 1 129 [Nie in die bron nie] ” 1 179 Beosmanland Boesmanland 2 190 [Nie in die bron nie] . 1 Afkortings Oorsig van afkortings gebruik. Afkorting Uitbreiding b.v. byvoorbeeld v.K. voor Kristus
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