^{1}

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For a graph , a function is called an edge product cordial labeling of G, if the induced vertex labeling function is defined by the product of the labels of the incident edges as such that the number of edges with label 1 and the number of edges with label 0 differ by at most 1 and the number of vertices with label 1 and the number of vertices with label 0 differ by at most 1. In this paper, we show that the graphs obtained by duplication of a vertex, duplication of a vertex by an edge or duplication of an edge by a vertex in a crown graph are edge product cordial. Moreover, we show that the graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is edge product cordial. We also show that the graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial.

We begin with a simple, finite, undirected graph

Definition 1. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition. If the domain of the mapping is the set of vertices, edges or both then the labeling is called a vertex labeling, an edge labeling or a total labeling.

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

We denote the number of vertices of G having label i under

The function f is called an edge product cordial labeling of G if

The concept of edge product cordial labeling was introduce by Vaidya and Barasara [

Definition 3. The graph

Definition 4. The helm

Definition 5. A gear graph is obtained from the wheel graph

Definition 6. The crown

Definition 7. The neighborhood of a vertex v of a graph is the set of all vertices adjacent to v. It is denoted by

Definition 8. Duplication of a vertex of the graph G is the graph

Definition 9. Duplication of a vertex

produces a new graph

The concept of duplication of vertex by edge was introduce by Vaidya and Barasara [

Definition 10. Duplication of an edge

The concept of duplication of edge by vertex was introduce by Vaidya and Dani [

Theorem 1. The graph obtained by duplication of an arbitrary vertex of the cycle in a crown graph is an edge product cordial graph.

Proof. Let

Let G be the graph obtained by duplication of the vertex

Thus

Now for

Case 1: When n is odd. For

In the view of above labeling pattern we have,

Case 2: When n is even. For

In the view of the above labeling pattern we have,

Thus, from both the cases we have

Hence, graph G admits edge product cordial labeling. Thus, G is an edge product cordial graph.

□Illustration 1. The graph obtained by duplication of an arbitrary vertex of the cycle

Theorem 2. The graph obtained by duplication of an arbitrary vertex of the cycle by a new edge in a crown graph is edge product cordial graph.

Proof. Let

Let G be the graph obtained by duplication of the vertex

In the view of the above labeling pattern we have,

Hence, graph G admits edge product cordial labeling. Thus, G is an edge product cordial graph.

□Illustration 2. The graph obtained by duplication of an arbitrary vertex of the cycle

Theorem 3. The graph obtained by duplication of an arbitrary edge of the cycle

Proof. Let

Let G be the graph obtained by duplication of an edge

In the view of above labeling pattern we have,

Hence, graph G admits edge product cordial labeling. Thus, G is an edge product cordial graph.

□Illustration 3. The graph obtained by duplication of an arbitrary edge of the cycle

Theorem 4. The graph obtained by duplication of each pendent vertex by a new vertex in a crown graph is edge product cordial graph.

Proof. Let

Let G be the graph obtained by duplication of each pendent vertex

Case 1: When n is odd, define

In the view of above labeling pattern we have,

Case 2: When n is even, define

In the view of above labeling pattern we have,

Thus, from both the cases we have

□Illustration 4. The graph obtained by duplication of each pendent vertex by a new vertex in a crown graph is edge product cordial graph as shown in

Theorem 5. The graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is an edge product cordial graph.

Proof. Let

Let G be the graph obtained from

Define

In the view of the above labeling pattern we have,

□Illustration 5. The graph obtained by duplication of each vertex of degree three by an edge in a gear graph is an edge product cordial graph as shown in

Theorem 6. The graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial graph.

Proof. Let v be the apex vertex and

Let G be the graph obtained from

Case 1: When n is odd, define

In the view of above labeling pattern we have,

Case 2: When n is even, define

In the view of above labeling pattern we have,

Thus, from both the cases we have

□Illustration 6. The graph obtained by duplication of each pendent vertex by a new vertex in a helm graph is edge product cordial graph as shown in

We have derived six results for edge product cordial related to crown graph, gear graph and helm graph in the context of duplication of various graph elements. Similar pro- blem can be discussed for other graph family for edge product cordial labeling.

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 Shah, P.D. (2016) Some Edge Product Cordial Graphs in the Context of Duplication of Some Graph Elements. Open Journal of Discrete Mathematics, 6, 248-258. http://dx.doi.org/10.4236/ojdm.2016.64021