^{1}

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For a graph having no isolated vertex, a function 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.

We begin with simple, finite, undirected graph

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 [

Definition 2. For a graph G, the edge labeling function is defined as

Let

f is called an edge product cordial labeling of graph G if

Vaidya and Barasara [

Definition 3. The wheel

Definition 4. The helm

Definition 5. The closed helm

Definition 6. The web

Definition 7. The Closed Web graph

Definition 8. [

Definition 9. [

Definition 10. [

Definition 11. [

Definition 12. The Flower graph

Theorem 1. Closed web graph

Proof. Let

Case 1: If n is odd 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

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

From both the cases

Theorem 2. Lotus inside circle

Proof. Let

So,

Theorem 3. Sunflower graph

Proof. Let

In view of the above defined labeling pattern we have,

Illustration 1. Graph

Theorem 4. The graph obtained from duplication of each of the vertices

Proof. Let

Case 1: If n is odd, define the mapping

of 6n edges. So in this context, the edges with label 0 will give rise at least

vertices with label 0 and at most

Therefore

Case 2: If n is even, define the mapping

In view of the above defined labeling pattern we have,

So

Illustration 2. Graph G obtained from

Theorem 5. The graph obtained from duplication of each of the vertices

Proof. Let

Case 1: If n is odd, define the mapping

In view of the above defined labeling pattern we have,

and

Case 2: If n is even, define

In view of the above defined labeling pattern we have,

and

From both the cases

Illustration 3. Graph G obtained from

Theorem 6. The graph obtained by duplication of each of the vertices in the sunflower graph

Proof. Let

Let

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

label 1 out of

Theorem 7. The graph obtained by subdividing the edges

Proof. Let

Case 1: If n is odd, define

In view of the above defined labeling pattern we have,

Case 2: If n is even, define

In view of the above defined labeling pattern we have,

From both the cases

Illustration 4. Graph G obtained from

Theorem 8. The graph obtained by flower graph

Proof. The Flower graph

each pendant vertex to the apex vertex of the helm

Let

Case 1: If n is even, define the mapping

In view of the above defined labeling pattern we have,

Case 2: If n is odd, define the mapping

In view of the above defined labeling pattern we have,

From both the cases

Illustration 5. Graph G obtained from

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.

The authors are highly thankful to the anonymous referee for valuable comments and constructive suggestions. The first author is thankful to the University Grant Commission, India for supporting him with Minor Research Project under No. F. 47-903/14 (WRO) dated 11th March, 2015.

Prajapati, U.M. and Patel, N.B. (2016) Edge Product Cordial Labeling of Some Cycle Related Graphs. Open Journal of Discrete Mathematics, 6, 268-278. http://dx.doi.org/10.4236/ojdm.2016.64023