Edge Product Cordial Labeling of Some Cycle Related Graphs

For a graph ( ) ( ) ( ) , G V G E G = having no isolated vertex, a function ( ) { } : ,1 f E G → 0 is called an edge product cordial labeling of graph G, if the induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and the number of edges with label 1 differ by at most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at most 1. In this paper, we discuss edge product cordial labeling for some cycle related graphs.


Introduction
We begin with simple, finite, undirected graph ( ) ( ) ( ) ( ) E G denote the number of vertices and edges respectively.For all other terminology, we follow Gross [1].In the present investigations, n C denotes cycle graph with n vertices.We will give brief summary of definitions which are useful for the present work.
Definition 1.A graph labeling is an assignment of integers to the vertices or edges or both subject to the certain conditions.If the domain of the mapping is the set of vertices (or edges) then the labeling is called a vertex (or an edge) labeling.
For an extensive survey on graph labeling and bibliography references, we refer to Gallian [2].Definition 2. For a graph G, the edge labeling function is defined as and induced vertex labeling function , , , n e e e  are all the edges incident to the vertex v then ( ) ( ) ( ) ( ) ( ) f e i be the number of edges of G having label i under f for 1, 2 i = .
f is called an edge product cordial labeling of graph G if ( ) ( ) ( ) ( ) A graph G is called edge product cordial if it admits an edge product cordial labeling.
Vaidya and Barasara [3] introduced the concept of the edge product cordial labeling as an edge analogue of the product cordial labeling.Definition 3. The wheel n W ( ) . In other words, v′ is said to be a duplication of v if all the vertices which are adjacent to v in G are also adjacent to v′ in G′ .Definition 11. [5] Duplication of a vertex k v by a new edge Definition 12.The Flower graph n Fl is the graph obtained from a helm n H by joining each pendant vertex to the apex of the helm n H .   ) ( ) ( )

Main Result
vertices with label 0 and at most vertices with label 1 out of 3 1 n + vertices.Therefore ( ) ( ) CWb is not an edge product cordial graph for odd n.Case 2: If n is even then in order to satisfy the edge condition for edge product cordial graph it is essential to assign label 0 to 3n edges out of 6n edges.So in this context, the edges with label 0 will give rise at least 3 2 2 n + vertices with label 0 and at most 3 1 2 n − vertices with label 1 out of 3 1 n + vertices.
From both the cases ( ) ( ) , , , respectively.Form the cycle by creating the edges We are trying to define the mapping In order to satisfy an edge condition for edge product cordial graph it is essential to assign label 0 to 2n edges out of 4n edges.So in this context, the edges with label 0 will give rise to at least 2 n + vertices with label 0 and at most So, n LC is not an edge product cordial graph.( ) In view of the above defined labeling pattern we have, ( ) ( ) . Thus ( ) ( ) ( ) ( ) ( ) ( ) 3 1 + .We consider the following two cases: Case 1: If n is odd, define the mapping in order to satisfy edge condition for edge product cordial graph it is essential to assign label 0 to 3n edges out of 6n edges.So in this context, the edges with label 0 will give rise at least  ( ) ≥ .So the duplication of vertex i w by vertex i u in sunflower graph is not edge product cordial for odd n.Case 2: If n is even, define the mapping as follows: ( ) In view of the above defined labeling pattern we have, , , , ,  ,  , , 1 , , , , , , , ( ) ( ) . Thus ( ) ( ) ( ) ( ) ≤ .Thus g admits an edge product cordial labeling on G. So, G is an edge product cordial for even n.( ) 4 1 + .We consider the following two cases: Case 1: If n is odd, define the mapping as follows: In view of the above defined labeling pattern we have, ( ) as follows: In view of the above defined labeling pattern we have, , ,  , , , ,  , , , ,  , , , , .
( ) ( ) From both the cases ( ) ( ) ( ) ( ) , , , , , , , , We are trying to define In order to satisfy the edge condition for edge product cordial graph, it is essential to assign label 0 to 6n edges out of 12n edges.So in this context, the edge with label 0 will give rise at least 4 1 1 2 vertices with label 0 and at most 4 1 2 So the graph G is not an edge product cordial graph.

Concluding Remarks
We investigated eight results on the edge product cordial labeling of various graph generated by a cycle.Similar problem can be discussed for other graph families.
no isolated vertex where ( ) V G and ( ) E G denote the vertex set and the edge set respectively, the number of vertices of G having label i under * f and the cycle by creating the edges

Theorem 3 ..
Sunflower graph n SF is an edge product cordial graph for 3 n ≥ .Proof.Let n SF be the sunflower graph, where 0 v is the apex vertex, Let 1 2 3 , , , , n e e e e  be the consecutive rim edges of the n W . 1 2 , , , n e e e ′ ′ ′  are the corresponding edges joining apex vertex 0 v to the vertices 1 2 , , , n v v v  of the cycle.Let for each 1 i n ≤ ≤ , l i e be the edges joining i w to i v and r i e be the edges joining i w to

Figure 1 .
Figure 1.SF6 and its edge product cordial labeling

Illustration 2 .
Graph G obtained from 6 SF by duplication of each of the vertices u u u and its edge product cordial labeling is shown in Figure 2. Theorem 5.The graph obtained from duplication of each of the vertices i w for 1, 2, , i n =  by a new edges i f in the sunflower graph n SF is an edge product cordial graph.

Figure 2 .
Figure 2. G and its edge product cordial labeling.

5 SF by duplication of each of the vertices 1
, g admits an edge product cordial labeling on G.So the graph G is an edge product cordial graph.Illustration 3. Graph G obtained from f f and its edge product cordial labeling is shown in Figure3.Theorem 6.The graph obtained by duplication of each of the vertices in the sunflower graph n SF is not an edge product cordial graph.Proof.Let n SF be the sunflower graph, where 0 v is the apex vertex, 1 2 , , , n v v v  be the consecutive vertices of the cycle n C and 1 2 , , , n w w w  be the additional vertices where i w is joined to i v and

Figure 3 .
Figure 3. G and its edge product cordial labeling.
g admits an edge product cordial on G.So the graph G is an edge product cordial graph.Illustration 4. Graph G obtained from 8 SF by subdividing the edges i i w v and product cordial labeling is shown in Figure 4. Theorem 8.The graph obtained by flower graph n Fl by adding n pendant vertices to the apex vertex 0 v is an edge product cordial graph.Proof.The Flower graph n Fl is the graph obtained from a helm n H by joining

Figure 4 .
Figure 4. G and its edge product cordial labeling.


In view of the above defined labeling pattern we have, Figure 5.

Figure 5 .
Figure 5. G and its edge product cordial labeling.
n > is the graph obtained by adding a new vertex joining to each of the vertices of n C .The new vertex is called the apex vertex and the vertices corresponding to n C are called rim vertices of n W .The edges joining rim vertices are called rim edges.Definition 4. The helm n H is the graph obtained from a wheel n W by attaching a pendant edge to each of the rim vertices.Definition 5.The closed helm n CH is the graph obtained from a helm n H by joining each pendant vertex to form a cycle.The cycle obtained in this manner is called an outer cycle.Definition 6.The web n Wb is the graph obtained by joining the pendant vertices of a helm n H to form a cycle and then adding a pendant edge to each of the vertices of the outer cycle.Definition 7. The Closed Web graph n CWb is the graph obtained from a web graph n Wb by joining each of the outer pendent vertices consecutively to form a cycle.Definition 8. [4] The Sunflower graph n SF is the graph obtained by taking a so Thus f admits an edge product cordial on fbe the edges joining i u to n by a vertex in the sunflower graph each pendant vertex to the apex vertex of the helm