Odd Graceful Labeling of Some New Graphs

In this work some new odd graceful graphs are investigated. We prove that the graph obtained by joining two copies of even cycle Cn with path Pk and two copies of even cycle Cn sharing a common edge are odd graceful graphs. In addition to this we derive that the splitting graph of K1, n as well as the tensor product of K1,n and P2 admits odd graceful labeling.


Introduction
We begin with simple, finite and undirected graph ( ( )

, ( )) G V G E G 
with p vertices and q edges.For standard terminology and notations we follow (Harary F., 1972).We will provide brief summary of definitions and other information which serve as prerequisites for the present investigations.Definition 1.1 If the vertices are assigned values subject to certain conditions then it is known as graph labeling.Labeled graphs have many diversified applications in coding theory particularly for missile guidance codes, design of good radar type codes and convolution codes with optimal autocorrelation properties.A systematic study of variety of applications of graph labeling is carried out by (Bloom G.S. and Golomb S.W., 1977).For detailed survey on graph labeling and related results we refer to (Gallian J.A., 2009).Most of the graph labeling schemes found their origin with graceful labeling which was introduced by (Rosa A., 1967) .
which admits graceful labeling is called a graceful graph.
Many researchers have carried out significant work on graceful labeling.For e.g.In (Golomb S.W.,1972) it has been proved that complete graph K n is not graceful for 5 n  .In (Drake A and Redl T.A., 2006) the non graceful Eulerian graphs are enumerated.The famous Ringel-Kotzig graceful tree conjecture and illustrious work in (Kotzig A., 1973) brought a tide of labeling problems having graceful theme.The present work is targeted to discuss one such labeling known as odd graceful labeling which is defined as follows.Definition 1.3 A graph G = (V (G),E(G)) with p vertices and q edges is said to admit odd graceful labeling if is injective and the induced function A graph which admits odd graceful labeling is called an odd graceful graph.(Gnanajothi R.B.,1991) introduced the concept of odd graceful graphs and she has proved many results on this newly defined concept.(Kathiresan K.M.,2008) has discussed odd gracefulness of ladders and graphs obtained from them by subdividing each step exactly once.(Sekar C.,2002) has proved that the splitting graph of path P n and the splitting graph of even cycle C n are odd graceful graphs.
Definition 1.4 For a graph G the splitting graph is obtained by adding to each vertex v, a new vertex ' v so that ' v is adjacent to every vertex that is adjacent to v in G.
and the edge set Here we prove that the graph obtained by joining two copies of even cycles by path P k , two copies of even cycles sharing a common edge, splitting graph of 1,n K are graceful graphs.We also show that tensor product of 1,n K and P 2 admits odd graceful labeling.

Main Results
Theorem 2.1 The graph obtained by joining two copies of cycle C n of even order with the path P k admits odd graceful labeling.
Proof: Let 1 2 , ,..., n v v v be the vertices of cycle C n and 1 2 , ,..., k u u u be the vertices of path P k .Consider two copies of C n of even order.Let G be the graph obtained by connecting two copies of C n with path be the vertices of G and these vertices form a spanning path in G.In this spanning path the vertex n v is the vertex common to the first copy of C n and path P k as well as the vertex is the vertex common to the second copy of C n and path P k .Define : ( ) {0,1, 2,..., 2 1} f V G q   as follows.
For 1 In accordance with the above labeling pattern the graph under consideration admits odd graceful labeling.
Above defined labeling pattern exhausts all possibilities and in each case the graph under consideration admits graceful labeling.
In view of the above defined labeling pattern G admits odd graceful labeling.u u u u  be the vertices of star 1,n K , with u 1 be the apex vertex.Let v 1 ,v 2 be the vertices of The above defined function f provides graceful labeling for tensor product of K 1,n and path P 2 .That is,

Concluding Remarks
Gracefulness and odd gracefulness of a graph are two entirely different concepts.A graph may possess one or both of these or neither.In the present work we investigate four new families of odd graceful graphs.To investigate similar results for other graph families and in the context of different labeling techniques is an open area of research.
and its odd graceful labeling is shown.

Figure 2 .
Figure 2. Odd graceful labeling of two copies of cycle C 8 sharing a common edge.

Figure 3 .
Figure 3. Odd graceful labeling of the splitting graph of K 1,4 .

Figure 4 .
Figure 4. Odd graceful labeling of the tensor product of K 1,4 and P 2 .
Illustration 2.2 Consider the graph obtained by attaching two copies of C 10 by P 5 .The labeling pattern is as shown in Fig 1. Theorem 2.3 Two copies of even cycle C n sharing a common edge is an odd graceful graph.
Illustration 2.4 Consider two copies of cycle C 8 sharing a common edge.The labeling pattern is as shown in Fig 2. Theorem 2.5 Splitting graph of a star admits odd graceful labeling.