Applications and Properties of Unique Coloring of Graphs

A. Sri Krishna Chaitanya. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 6, Issue 9,(Part-4) Sep.2016, pp.65-67
www.ijera.com 65|P a g e
Applications and Properties of Unique Coloring of Graphs
A. Sri Krishna Chaitanya1
Dr. R. Sivaprasad2
1
A. Sri Krishna Chaitanya, Research Scholar, Department of Mathematics, Sri Krishnadevaraya University,
Anantapur Dist., A.P.
2
Dr. R. Sivaprasad, Professor, Department of Mathematics, Sri Krishnadevaraya University, Anantapur Dist.,
A.P.
ABSTRACT
This paper studies the concepts of origin of uniquely colorable graphs, general results about unique vertex
colorings, assorted results about uniquely colorable graphs, complexity results for unique coloring
Mathematics Subject Classification 2000: 05CXX, 05C15, 05C20, 37E25.
Keywords: complete graph. bipartite graph, k-chromatic graph, adjacent vertices, maximal planar graphs.
I. ORIGIN OF UNIQUE COLORING
The origin of unique coloring appears to
have been, perhaps surprisingly, in the field of
psychology. There the problem of a signed graph
was introduced, together with a coloring of signed
graphs, to model a problem in that field [4]. A
signed graphS is a ordered pair (G;), where G is an
undirected graph and is a function: E(G)→{-1,1}.
These signed graphs are used in psychology to
model the idea of clusterings. From there the idea
of colorings and unique colorings a signed graph,
closely related to the normal notion of coloring a
graph arose in a 1968 paper of Cartwright and
Harary [31]A coloring cof a signed graph is a
function from the vertex set of G to{1; 2,….k}
having the property that if x and y are two
adjacentvertices in G, then
1) If ({x;y}) = 1 then c(x) = c(y).
2) If ({x;y}) = 1 then c(x) ≠ c(y).
As usual the set {c-1
({i}) : i {1; 2,….k}}
defines a partition of the vertices ofS into color
classes. This paper of Cartwright and Harary, as
well as a 1967 paper of Gleason and Cartwright
[31], established conditions for a signed graph to
have a coloring, and introduced the notion of a
unique coloring of a signed graph. To wit, a signed
graph S is uniquely colorable if there is exactly one
partition of S into color classes. Both papers gave
fairly simple criterion for a signed graph to be
uniquely colorable. In addition introduced the
notion of unique coloring of a “normal" (unsigned)
graph G, which is the topic of interest in this thesis.
Under the usual notion of a coloring c of a
graph G being a function from the set of vertices to
a set of integers (colors) having the property that
adjacent vertices receive a different assignment
under c, Cartwright and Harary defined a graph G
to be uniquely colorable if either G is complete or
G has a unique partition of the vertices of G into t
<|V (G)| color classes. In this same paper, they
showed that if G has a unique coloring with say t
colors, then, in fact t = (G), where (G) is the
chromatic number of G, that is, the smallest
positive integer s for which there is a coloring of G
using exactly scolors.
1.2 General Results about Unique Vertex
Colorings
Necessary Conditions for a Graph to be
uniquely Colorable
To warm up our understandings of unique
coloring we mention some easy necessary
consequences of a graph being uniquely vertex-
colorable. The first is that the number of colors
used in a unique coloring is unique and equals the
chromatic number of G.
Proposition1.2.1. (Cartwright, Harary) IfGhas a
unique coloring with t colors then t = (G).
Proof: We may assume that Gis not the complete
graph on (G) vertices. Clearly(G) <t < |V(G)|, since
a unique coloring is also a proper coloring. If t
>(G) then |V(G)|>(G) and for any (G)-coloring c of
G, pick a set of vertices {x1;x2,….,} having the
property that c(xi) = i. There are at least t –x (G)
vertices in G other than {x1,…,} and these can be
assigned colors from {(G) + 1; (G) + 2,…,t}, to get
two distinct t-colorings of G.
By this proposition, we may say unambiguously
that G is uniquely vertex colorable, and mean that
G is uniquely vertex- (G)-colorable.
Let G be a graph and let c : V(G)→{1, 2, …, k} be a
unique vertex-k-coloring of G. For i, j {1, 2, …, k},
define Gi;j to be the subgraph of G induced by the
vertices which c assigns the colors i or j. A very
useful necessary condition for G tobe uniquely
vertex-k-colorable was noticed by Harary et. al. in
the following theorem which appears in [30].
RESEARCH ARTICLE OPEN ACCESS

Theorem 1.2.2. (Harary, Hedetniemi, Robinson,
1969) Ifc:V(G) →{1,2,…,k} is a unique vertex-k-
coloring of G, then for all i ≠ j, i, j {1, 2, …, k}, the
graph Gi;j is connected.
Proof: If someGi;jhad two or more components,
then by interchanging the colorsiand j in exactly
one of these components, we would arrive at a
valid coloring different than c.
Corollary1.2.3. Letcbe a unique vertex-k-coloring
ofG, letxbe a vertex inV(G)and let i {1,….,k}. If i ≠
c(x) then there is a vertex yV (G) such that x is
adjacent to y and c(y) = i. In particular, every
vertex of G has degree at least k1.
Proof: LetuV(G), letcbe a unique vertex-k-coloring
ofGand letibe a colordifferent from c(x). By
Theorem 1.2.2., Gi;c(x) is a connected graph, and in
particular, x is not an isolated vertex in Gi;c(x)
because Proposition 1.2.1 insures that some vertex
receives the color i. Since there are k1 other colors
besides c(x), the minimum degree of G must be at
least k1. This completes the proof of Corollary
1.2.3.
Corollary 1.2.4.(Harary et al.) IfGis a uniquely
vertex-k-colorable, thenGhas at least (kedges.
Proof: LetVibe the set of vertices coloredi.
Theorem 1.2.2 insures that for 1<i < j<k, the graph
Gi;j with vertex set ViVj is connected. Thus |E(Gi;j)|
> |Vi|+|Vj|-1. Summing this inequality over all
pairsi≠j, we have that
|E(G)|)which isthe desired result.
Corollary1.2.5.(Geller, Chartrand) IfGis a
uniquely vertex-4-colorable simpleplanar graph,
then any drawing of the graph G is a triangulation.
Moreover, for i ≠ j and i, j{1; 2; 3; 4}, each
subgraph Gi;j is a tree.
Proof: By Euler's formula|E(G)| <3|V(G)|6, and
from Corollary 1.2.4,|E(G)|>3|V (G)|6, so |E(G)| =
3|V (G)|6. This implies that any drawing of G must
be a triangulation. It also implies that equality
holds throughout in the proof of the Corollary
1.2.4, so . Since Gi,j is connected, it follows that Gi,j
is a tree. This completes the proof of the corollary.
II. ASSORTED RESULTS ABOUT
UNIQUELY COLORABLE
GRAPHS
A function : V (G) → V ( ) is said to be
a homomorphism of the graph G into the graph
if it preserves adjacency of vertices, that is, if {x,y}
E(G) implies {(x);(y)}E( ). If it is true that for
every pair of vertices , isadjacent to in
G’
if and only if there is a pair x, y of adjacent
vertices in G such that (x) = and (y) = , then
is said to be a homomorphism of G onto , and
is said to be a homomorphism image of G. The
following propositions appear in [32].
Proposition 2.1.1. IfGis uniquely vertex-k-
colorable andHis a homomorphism image of G
such that (H) = k, then H is uniquely vertex-k-
colorable .
Proposition 2.1.2. IfGis uniquely vertex-k-
colorable thenGis(k-1)-connected.
Proof: LetAbe a set with|A| <k-2, letcbe a unique
vertex-k-coloring ofG,and let x, y V(G) A. There
are two distinct colors i, j {1,….,k} such that
novertex of A has a vertex colored i or j by c.
Therefore, . By Corollary 1.2.3, there are vertices
ux and uy such that x is adjacent to ux, y is adjacent
to uy and c(ux) = c(uy) = i. Since Gi;j is connected,
there is a path P in Gi,j joining uxto uy and thus
there is a path in GA joining x and y. Thus, G A is
connected. This completes the proof of the
proposition.
III. COMPLEXITY RESULTS FOR
UNIQUE COLORING
The following proposition is obvious.
Proposition 3.1.1. A graph is uniquely vertex-1-
colorable if and only if it consists ofisolated
vertices. A graph is uniquely vertex-2-colorable if
and only if it is a connected bipartite graph.
Beyond this there is not much hope of finding a
“good" characterization of arbitrary uniquely
vertex-k-colorable graphs when k>3 because of the
following complexity results contained in or
implied by the work of Dailey in 1981 [33]
Theorem 3.1.2. The following decision problems
are NP-Complete:
1. Given a graph G and a vertex-k-coloring c of
G, is there a vertex-k-coloring of G that is
not equivalent to c?
2. Given an integer k and a graph G, does G have
either 0 or at least 2 vertex-k-colorings?
The result of Dailey probably dooms any
possibility of a polynomial time algorithm for
problems 1) or 2) above. In [31], the authors pose
the question of whether there is a polynomial time
algorithm for deciding whether a given planar
graph is uniquely vertex-3-colorable. This problem
is still open as far as this author knows.
G
G
G
yx , x y
x y
G
G
c

IV. A SUFFICIENT CONDITION FOR
DETERMINING UNIQUE VERTEX-
K-COLORABILITY
The following sufficient condition for a graph to be
uniquely vertex-k-colorable was given by Bollobas
in [14].
Theorem 4.1.1. Letkbe an integer greater than
one, letGbe a vertex-k-colorablegraph on n
vertices, and let (G) denote the minimum degree of
G. If then G is uniquely vertex-k-colorable .
Moreover, if G has a vertex-k-coloring in which Gi;j
is connected for every 1 <i < j <k, and, then G is
uniquely vertex-k-colorable. These results are best
possible.
This was generalized by Dmitriev
according to a review of [15]. As we can see, this
condition will apply only to very dense graphs.
REFERENCES
[1]. Appel, K. and W. Haken(1977). Every
planar map is four colorable. Part I.
Discharging Illinois Journal of
Mathematics21, 429-490.
[2]. Bodlaender, H.L. 1991. “On the
complexity of some coloring games”,
Lecture Notes in Computer Science. No.
484, pp. 30-40.
[3]. Bondy, J.A. and Murty, U.S.R., 1985.
Graph Theory with Applications, (Elsevier
Science Publishing Co., Inc.).
[4]. F. Harary. Conditional colorability in
graphs. Appears in Graphs and
Applications: Proceedings of the First
Colorado Symposium on Graph Theory.
John Wiley and Sons Inc., New York,
1985.
[5]. F. Harary. Graph Theory. Addison-
Wesley, Massachusetts, 1969.
[6]. F. Harary and K. Fraughnaugh (Jones).
Conditional colorability ii:bipartite
variations. Congr. Numer.,50:205-218,
1985.
[7]. F. Harary and K. Fraughnaugh (Jones).
Degree conditional bipartition numbers in
graphs. Congr. Numer., 55:39-50, 1986.

Applications and Properties of Unique Coloring of Graphs

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