An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles

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10.5121/jgraphhoc.2010.2108 112
M. Ibrahim Moussa
Faculty of Computers & Information, Benha University, Benha, Egypt
moussa_6060@yahoo.com
ABSTRACT
In 1991, Gnanajothi [4] proved that the path graph nP with n vertex and 1n − edge is odd graceful, and
the cycle graph mC with m vertex and m edges is odd graceful if and only if m even, she proved the
cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph m nC P∪ when 4,6,8,10m =
and then we proved that the graph m nC P∪ is odd graceful if m is even. Finally, we described an
algorithm to label the vertices and the edges of the vertex set ( )m nV C P∪ and the edge set ( )m nE C P∪ .
KEY WORDS
Vertex labeling, edge labeling, odd graceful, Algorithms
1. INTRODUCTION
The study of graceful graphs and graceful labeling methods was introduced by Rosa [1]. Rosa
defined a - valuation of a graphG withq edges an injection from the vertices of G to the set
{ }0,1,2 q… such that when each edgeuv is assigned the label ( ) ( ) ,f u f v− the resulting edges are
distinct. - Valuation is a function that produces graceful labeling. However, the term graceful
labeling was not used until Golomb studied such labeling several years later [2]. A graph G of
size q is odd-graceful, if there is an injection f from ( )V G to { }0,1,2 2 1q… − such that, when
each edgeuv is assigned the label or weight ( ) ( ) ,f u f v− the resulting edge labels are
{ }1,3,5 2 1 .q… − This definition was introduced by Gnanajothi [4] in 1991. In 1991, Gnanajothi
[4] proved the graph 2C Km × is odd-graceful if and only if m even. She also proved that the
graph obtained from 2nP P× by deleting an edge that joins to end points of the nP paths and this
last graph knew as the ladder graph. She proved that every graph with an odd cycle is not odd
graceful. She also proved the following graphs are odd graceful: ;n mP C if and only if m is even
and the disjoint union of copies of 4C . In 2000, Kathiresan [6] used the notation ;mnP to denote
the graph obtained by identifying the end points of m paths each one has length n . In 1997
Eldergill [5] generalized Gnanajothi [4] result on stars by showing that the graph obtained by
joining one end point from each of any odd number of paths of equal length is odd graceful
graph. In 2002 Sekar [7] proved that the graphs; ;mnP when 2n ≥ and m is odd, 2;mP and
m 2, 4;mP and m 2, ;mnP when n and m are even and n 4 and m 4, 4 1;4 2 ,r rP + +
4 1;4 ,r rP − and all n-polygonal snakes with n even are odd graceful. In 2009 Moussa [9]
presented some algorithms to prove for all 2m≥ the following graphs 4 1;m 1,2,3, =rP r− and

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113
4 1;m =1,2,r rP + are odd graceful. He presented algorithm which show that the closed spider
graph and the graphs obtained by joining one or two paths mP to each vertex of the path nP are
odd graceful. He used the cycle representation and denoted £ -representation to present a
simple labeling graph algorithm, the cycle representation is similar to theπ -representation made
by Kotazig [3]. In 2009 Moussa and Bader [8] have presented the algorithms that showed the
graphs obtained by joining n pendant edges to each vertex of mC are odd graceful if and only if
m is even.
In this paper we show that nmC P∪ is odd graceful if m is even and 2n m> − , if the number of
edges in the cycle mC can be equally divisible by four, and 4n m> − for all other even value of m .
We first explicitly define an odd graceful labeling of 4 nC P∪ , 6 ,nC P∪ 8 10and,n nC P C P∪ ∪ and then,
using this odd graceful labeling, describe a recursive procedure to obtain an odd graceful
labeling of nmC P∪ . Finally; we present an algorithm for computing the odd graceful labeling of
the union of path and cycle graphs, we prove the correctness of the algorithms at the end of this
paper. The remainder of this paper is organized as follows. In section 2, illustrate the need of
graph labeling and we mention the existing variety types of labeling methods. In section we
give some assumptions and definitions related with the odd graceful labeling and graphs. In
section , we present and discuss the odd label of union of paths and cycle. Section is the
conclusion of this research.
2. RELATED WORK
The graph labeling serves as useful models for a broad range of applications such as: radar,
communications network, circuit design, coding theory, astronomy, x-ray, crystallography, data
base management and models for constraint programming over finite domains. J. Gallian in his
dynamic survey [11], he has collected everything on graph labeling, he observed that over
thousand papers have been studied and many kinds of graph labeling have been defined, viz.:
Graceful Labeling, Harmonious Labeling, Magic Labelings, balanced labeling, k -graceful
Labeling, -labeling, and Odd-Graceful Labelings. For further information about the graph
labeling, we advise the reader to refer to the brilliant dynamic survey on the subject [11].
3. ASSUMPTIONS AND DEFINITIONS.
Definition 1[4]
Let G be a finite simple graph, whose vertex set is denoted ( )V G , while ( )E G denotes its edge
set, the order of G is the cardinality ( )V Gn = and the size of G is the cardinality ( )GEq = . We
write ( )uv E G∈ if there is an edge connecting the vertices u and v inG . An odd graceful labeling
of a graph G is a one to one function : ( ) {0,1,2,...,2 1}.f V G q→ − Such that, when each edge uv is
assigned the label ( ) ( )( ) f ff uv u v−
∗
= the resulting edge labels are{ }0 , 1, 2 2 1q… − .
Definition 2[10]
A path in a graph is a sequence of vertices such that from each of its vertices there is an edge to
the next vertex in the sequence. The first vertex is called the start vertex and the last vertex is
called the end vertex. Both of them are called end or terminal vertices of the path. The other
vertices in the path are internal vertices. A cycle is a graph with an equal number of vertices and
edges whose vertices can be placed around a circle so that two vertices are adjacent if and only
if they appear consecutively along the circle. The graph has n or m vertex that is a path or a
cycle is denoted nP or mC , respectively. The union of two graphs ( )1 1 1,G V E= and ( )2 2 2,G V E= ,
written 1 2G G∪ , is the graph with vertex set 1 2 1 2( ) ( ) ( )V G G G GV V∪ = ∪ and the edge
set 1 2 1 2( ) ( ) ( )E G G G GE E∪ = ∪ .

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114
4. VARIATION OF ODD GRACEFUL LABELING
4.1. Odd gracefulness of 4 nC P∪
Theorem1 4 nC P∪ is odd graceful for every integer 2n > .
Proof
Let 4 1 2 3 4( ) { , , , }V C u u u u= , 1{ }( ) ,...,n nV v vP = where 4( )V C is the vertex set of the cycle 4C and ( )nV P
is the vertex set of the path nP , and 3q n= + , see Fig.1. For every vertex iu and iv , the odd
graceful labeling functions ( )iuf and ( )ivf respectively as follows
{
1 2 3 4
2 6
( ) 0 , ( ) 2 1 , ( ) 2 , ( ) 2 5 a n d
( )
i i o d d
i
q i i e v e n
f u f u q f u f u q
f v
− −
= = − = = −
=
The edge labeling function
*
f defined as follows:
1 2 2 3
* * *
3 4 4 1
*
1
( ) 2 1, ( ) 2 3, ( ) 2 7, ( ) 2 5, and
( ) 2 2 7 1, 2, ..., 1i i
f u q f u q f u q f u q
f v v q i i n
u u u u
∗
+
= − = − = − = −
= − − = −
Figure1shows the method labeling of the graph 4 nC P∪ this complete the proof.
24 22 20 18 16 14 0
31 27
23 19 15 11 7 3
31 27
21 17 13 9 5 1 29 25
1 3 5 7 9 11 13
Figure 1.
Theorem2 6 nC P∪ is odd graceful for every integer 2n > .
Proof
Let 6 1 2 3 4 5 6( ) { , , , }, ,V C u u u u u u= , 1{ }( ) , ...,n nV v vP = where 6( )V C is the vertex set of the cycle 6C
and ( )nV P is the vertex set of the path nP , and 5q n += , see Fig. 2. For every
vertex iu and iv , the odd graceful labeling functions ( )iuf and ( )ivf respectively as follows:
1 2 3 4 5 6
2 3
1
2 1 2
( ) 0 , ( ) 2 1, ( ) 2 3 , ( ) 4 , ( ) 2 9
a n d ( )
( ) 2 ,
i
i i o d d
i i
q i i e v e n
f u f u q f u q f u f u q
f v
f u
+ ≤
=
− +
= = − = − = = −
=
=
The edge labeling function *
f defined as follows:
1 2 2 3
* * *
3 4 4 5
* *
5 6 6 1
( ) 2 1, ( ) 2 3, ( ) 2 5, ( ) 2 7,
( ) 2 13, and ( ) 2 9
u u u u u u u u
u u u u
f q f q f q f q
f q f q
∗
= − = − = − = −
= − = −

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115
18 16 14 12 0
27 19
11 27 19
17 13 9 7 5 3 1 25 15
2 4
1 5 7 9 11 23 21
Figure 2. 25
Theorem 3 8 nC P∪ is odd graceful if and only if n 7.
Proof:
For every vertex iu and iv in 8( )nV P C∪ , we defined the odd graceful labeling
functions ( )iuf and ( )ivf respectively as follows
( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 5 6 7
0, 2 1, 2, 2 3, 4, 2 5, 6,u u q u u q u u q uf f f f f f f= = − = = − = = − =
( )8
2 1 4 2
2 1 4 4
2 1 3 , a n d ( )
i i o d d
q i
q i i e v e n
iu q f vf − =
− − ≤
= − =
And the function *
f is defined as follows:
1 2 2 3 3 4 4 5
5 6 6 7 7 8 8 1
* * * *
* * * *
( ) 2 1, ( ) 2 3, ( ) 2 5 , ( ) 2 7 ,
( ) 2 9 , ( ) 2 11, ) 2 1 9 , ( ) 2 13
( ) 2 1 5, ( ) 2 17 , ( ) 2 2 15 , 3, ..., 8
* * *
1 2 2 3 1
(
u q f u q f u q f u q
u q f u q f q f u q
f v v q f v v q f v v q i i qi i
f u u u u
f u u u u u
= − = − = − = −
= − = − = − = −
= − = − = − − = −+
The labeling of the graph 8 12C P∪ is indicated by Fig.3.
0
24 2 18 14
37
23 17 13 9 5 1
21 15 11 7 3
1 3 5 7 9 11 35
Figure 3.
Theorem 4 10 nC P∪ is odd graceful for every integer n .
Proof:
For every vertex iu and iv , we defined the odd graceful labeling functions ( )iuf and
( )ivf respectively as follows:

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116
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 3 4 5 6
7 8 9 10
0, 2 1, 2, 2 3, 4, 2 5,
6, 2 7, 8, 2 17, and
u u q u u q u u q
u u q u u q
f f f f f f
f f f f
= = − = = − = = −
= = − = = −
2 5
1 , 3
2 1 6
( )
i i o d d
i i
q i i e v e n
if v
+ ≤
=
− −
=
So we obtain all the edge labels and the function *
f is defined as follows:
1 2 2 3 3 4 4 5
* * * *
( ) 2 1, ( ) 2 3, ( ) 2 5, ( ) 2 7u u u u u u u uq f q f q f qf = − = − = − = −
4.5. Odd gracefulness of m nC P∪
Theorem 5
Let k is a given integer and 2km = , the graph nmC P∪ is odd graceful for every 2n m> − , k is
even, if k is odd number the graph nmC P∪ is odd graceful for every 4n m> − .
Proof:
Let 1( ) { ,..., }m mV C u u= , 1( ) { , ..., }n nV P v v= , where ( )mV C is the vertex set of the cycle mC and
( )nV P is the vertex set of the path nP , and 1mq n + −= . For every vertex iu and iv , we
defined the odd graceful labeling functions ( )iuf and ( )ivf respectively as follows:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 3 4
5 6 9
0, 2 1, 2, 2 3,
4, 2 5, 8,..., 2 (2 3)
m
q q
q q m
u u u u
u u u u
f f f f
f f f f
= = − = = −
= = − = = − −
If the value 2 ,k km = is odd number, the vertex 2v would be labeled 2
2 2 2( ) q mf v − +=
which decreased by two at every new value 4, 6, ..., 2ki −= , this means that
2( ) ( ) 2 2 2 ( 4)iif v f v q m i−= − = − − − , and
2
1 , 3 , ..., 2
2 2 ( 4 )
( )i
i k i o d d
i i k
q m i i e v e n
f v
+ ≤
= −
− − −
=
If 2 ,k km= is even number, the vertex 2v would be labeled 2
2 2 2( ) q mf v − += which decreased by
two at every new value 4,6,..., 2ki −= . For 2i k −= the label value is 2
2 2 6( )k
q m kf v −
− + −= while
the label value of the vertex kv is four out of the value 2( )kf v − , this means that
2 4( ) ( ) 2 2 2i k kf v f v q m i−= = − = − + − , and
2 2 4 2 , 4 , ..., - 2
2 2 2
( ) q m ii
i i o d d
i k
q m i k i e v e n
f v − + =−
− + − ≤
=

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117
The function *
f induces the edge labels of the cycle mC as the following:
1 2 2 3 3 4
* * *
( ) 2 1, ( ) 2 3, ( ) 2 5, . . . ,u q f u q f u qf u u u= − = − = −
1 1
* *
) 2 3 5 , ( ) 2 2 3( m m mf q m f u q mu u u− = − + = − + .
Function *
f induces the edge labels of the path as follows:
1 2 2 3 2 1
1 1
* * *
* *
( ) 2 2 1, ( ) 2 2 1, ..., ( ) 2 3 7,
( ) 2 3 3, ......., ) 2 3 1, ........,1(
k k
k k k k
v q m f v q m v q m
f v q m f q m
f v v v
v v v
f − −
− +
= − + = − − = − +
= − + = − +
There is a guarantee that each component in the given graph has odd graceful, the path graph is
odd graceful, the cycle graph with an even number of vertices is odd graceful (see [4]). We
have to prove that the vertex labels are distinct and all the edge labels are distinct odd
numbers { }1,3,5 2 1q… − . The edge labels of mC are numbered according to the decreasing
sequence 2 1, 2 3, ......q q− − . The edge labels of nP are numbered according to the decreasing
sequence *
1( ) 2 2 (2 1), 4,...,i if v v q i m i q m+ = − − − = − . The reader can easily find out, if i q m= − the last
edge label is equal one; this means that the edge labels take values in{ }2 1,2 3, ,1q q− − … . In order
to prevent any vertex in nP to share label with a vertex in mC , the difference between the largest
even label and the smallest even label in nP have to be more than the largest even label in mC ,
this leads to two cases:
Case I: if 2 ,m k k= is even then 21) 2 2( ( ) 2m nf u m n m− − > − > −
Case I: if 2 ,m k k= is odd then 21) 2( 2( ( ) 1) 4m nf u m n m− − − > − > − .
4.6. The proposed sequential algorithm
The union graph n mP C∪ has a vertex set ( ( ) ( )) n mn mV P V P V CC ∪=∪ with cardinalityn m+ and an
edge set ( ( ) ( )) n mn mE P E P E CC ∪=∪ with cardinality 1q m n= + − . Let the cycle mC is demonstrated
by listing the vertices and the edges in the order 1 1 2 2 1 1 1, , , ,..., , , , ,m m m mu e u e u e u e u− − . We name the
vertex mu ACTIVE vertex, the vertex mu is an endpoint of the edge 1me − , and we name the edge
1me − DOUBLE-JUMP edge. The path nP is demonstrated by listing the vertices and the edges
in the order 1 1 2 2 1 1, , , ,..., , ,n n nv e v e v e v− −
′ ′ ′ ′ . The algorithm has two passes; they can run in a
sequential or a parallel way and it works in a way similar to the above labeling in section 3.5. In
one pass, the algorithm labels the vertices and the edges in the cycle mC . For the other pass, it
labels the vertices and the edges of the path nP . At the beginning of the algorithm, we are
computing the odd label function for the ACTIVE vertex and the DOUBLE-JUMP edge. The
ACTIVE vertex has the odd graceful labeling function ( ) 2 (2 3)mf u q m= − − , the vertex mu has
the smallest odd label value between the vertices in the cycle .mC The DOUBLE-JUMP edge is
assigned the label function *
1( ) 2 3 5mf e q m− = − + . The given label to the ACTIVE vertex and
the DOUBLE-JUMP edge computed independently from other vertices or edges in the graph.
1. Number the ACTIVE vertex with the value ( ) 2 (2 3)mf u q m= − −
2. Number the DOUBLE-JUMP edge with the value *
1( ) 2 3 5mf e q m− = − +
Algorithm 1: Procedure Initialization
In the first pass, the algorithm starts at the vertex 1u , there are two main steps that can be
performed. These steps (in particular order) are: performing an action on the current vertex

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118
(referred to as "numbering" the vertex), number the current vertex with the value ( ) 01f u = ,
traversing to the left adjacent vertex 2u and number it with the value ( ) 2 12 qf u = − , and
traversing to the left adjacent vertex 3u and number it with the value ( )3 2f u = traversing to the
left adjacent vertex 4u and number it with the value ( ) 2 34 qf u = − , traversing to the left
adjacent vertex 5u and number it with the value ( )5 4f u = .. Thus the process is most easily
described through recursion. Finally, reach to the ACTIVE vertex which has the exception label
and number it with the value ( ) 2 (2 3)mf u q m= − − , the edge’s labeling induced by the absolute
value of the difference of the vertex’s labeling. To label the cycle mC odd graceful, perform the
following operations, starting with 1u :
1. Number the vertex 1u with the value ( ) 01f u =
2. For ( i = 3; i m-2; i += 2 )
( ) ( )2 2i if u f u −= +
3. For ( i = 2; i m-1; i += 2 )
( ) 12if u q i= − +
4. Number the ACTIVE vertexwith ( ) 2 (2 3).mf u q m= − −
5. Compute the edge labels by taking the absolute value of the difference of incident vertex
labels.
Algorithm2: Odd graceful labeling of mC
After the above process, the algorithm starts the second pass to label the vertices and edges of
the path component nP . Second pass starts at the edge 1 21 ( , )e v v′ = , its label value
is 1 2( )( ) mf e f u∗
−′ = , if the label value of the edge 1e′equals to the label value of the DOUBLE-
JUMP edge renumber it with the value ( ) ( ) 21 1f fe e= −
∗ ∗
′ ′ and number the vertex 1v with the
label value 1( ) 1,f v = traversing to the vertex 2v and number it with the value
2 1( ) ( ) 1f ef v ∗ ′= + . Traversing to the next incident edge 2e′ and number it with the
value ( ) ( ) 22 1f fe e= −
∗ ∗
′ ′ if the label value of the edge 2e′ equals to the label value of the
DOUBLE-JUMP edge renumber it with the value ( ) ( ) 22 2f fe e= −
∗ ∗
′ ′ , traverse to the next
vertex 3v which induces the label value 3 2 2( ) ( ) ( )f vf v f e∗
= ′− , otherwise traverse to the next
vertex 3v , without double subtracting for the label value of the edge 2e′ , and number it with the
value 3 2 2( ) ( ) ( )f vf v f e∗
= ′− , traverse to the next vertex 4v which induces the label value
4 3 3( ) ( ) ( )f vf v f e∗
= ′+ . Thus the process is most easily described through recursion again. To
label the path nP odd graceful labeling, perform the following operations, starting with the
edge 1 21 ( , )e v v′ = :
1. Number the vertex 1v with the value ( ) 11f v =
2. Number an auxiliary edge 0e′ with 0 ( )( ) mf e f u∗
′ =
3. For ( j = 1; j n-1; j += 1 )
3.1 Number the edge je′ with ( ) ( ) 21j jf fe e= −
∗ ∗
−
′ ′
3.2 If ( ( ) ( )1mjf f ee −
=
∗ ∗
′ ) Renumber the edge je′ with the value ( ) ( ) 2j jf fe e= −
∗ ∗
′ ′
3.3 Number the vertex 1jv + with the value 1
1 )( ) (( ) ( )j
j j jef v f f v+
+
∗
−′ +=
4. End For

! " #!#
119
Algorithm 3: Odd graceful labeling of nP
The algorithm is traversed exactly once for each vertex and edge in the graph n mP C∪ , since the
size of the graph equals q then at most O ( q ) time is spent in total labeling of the vertices and
edges, thus the total running time of the algorithm is O ( q ). The parallel algorithm for the odd
graceful labeling of the graph n mP C∪ , based on the above proposed sequential algorithm is
building easily. Since all the above three subroutine are independent and there is no reason to
sort their executing out, so they are to join up parallel in the same time point.
5. CONCLUSION
In this paper, we first explicitly defined an odd graceful labeling of 4 nC P∪ , 6 ,nC P∪
8 10and,n nC P C P∪ ∪ and then using this odd graceful labeling to have generalized results by
describing a recursive procedure to obtain an odd graceful labeling of nmC P∪ , if m is even and
2n m> − , if the number of edges in the cycle mC can be equally divisible by four,
and 4n m> − for all other even value of m . After we introduced a general form for labeling the
union of the paths and the cycles in odd graceful label, we described a sequential algorithm to
label the vertices and the edges of the graph n mP C∪ . The sequential algorithm runs in linear
with total running time equals O ( q ). The parallel version of the proposed algorithm, as we
showed, existed and it is described shortly.
REFERENCES
[1] A. Rosa (1967), On certain valuations of the vertices of a graph, Theory of Graphs (Internet
Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris 349-355.
[2] S. W. Golomb (1972) How to number a graph: graph theory and computing, R.C.Read,
ed.Academic Press: 23-37.
[3] A. Kotazig, On certain vertex valuation of finite graphs, Utilitas Math.4 (1973)261-290.
[4] R.B. Gnanajothi (1991), Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University,
India.
[5] P. Eldergill, Decomposition of the Complete Graph with an Even Number of Vertices, M. Sc.
Thesis, McMaster University, 1997.
[6] K. Kathiresan, Two classes of graceful graphs, Ars. Combin.55 (2000) 129-132.
[7] C. Sekar, Studies in Graph Theory, Ph. D. Thesis, Madurai Kamaraj University, 2002.
[8] M. I. Moussa & E. M. Badr “ODD GRACEFUL LABELINGS OF CROWN GRAPHS”
1stINTERNATIONAL CONFERENCE Computer Science from Algorithms to Applications
2009 Cairo, Egypt.
[9] MAHMOUD I. MOUSSA “Some Simple Algorithms for Some Odd Graceful Labeling Graphs”
Proceedings of the 9th WSEAS International Conference on APPLIED INFORMATICS AND
COMMUNICATIONS (AIC '09) August, 2009, Moscow, Russia.
[10] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Elsevier North-Holland, 1976.
[11] J. Gallian A dynamic survey of graph labeling, the electronic journal of combinatorics 16 (2009),
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