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A graph is said to be a product cordial graph if there exists a function with each edge assign the label , such that the number of vertices with label 0 and the number of vertices with label 1 differ atmost by 1, and the number of edges with label 0 and the number of edges with label 1 differ by atmost 1. We discuss the product cordial labeling of the graphs obtained by duplication of some graph elements of gear graph. Also, we derive some product cordial graphs obtained by vertex switching operation on gear graph.

We begin a simple, finite, undirected graph

Definition 1. If the vertices or edges or both of the graph are assigned values subject to certain conditions then it is known as vertex, edge, total labeling respectively.

For latest survey on graph labeling, we refer to Gallian [

Definition 2. A mapping

Definition 3. Given

Definition 4. A product cordial labeling of graph G with vertex set V is a function

The notion product cordial labeling was introduced by Sundaram, Ponraj and Soma- sundaram [

Definition 5. The neighborhood of a vertex v of a graph is the set of all vertices adjacent to v. It is denoted by

Definition 6. Duplication of a vertex of the graph G is the graph

Definition 7. Duplication of a vertex

Definition 8. Duplication of an edge

The notions of duplication of a vertex by a new edge and duplication of an edge by a new vertex were introduced by Vaidya and Barasara [

Definition 9. For a graph G and a vertex v of G, define a vertex switching

The notion vertex switching was introduced by Vaidya, Srivastav, Kaneria, and Kanani [

Definition 10. The graph

Definition 11. A gear graph is obtained from the wheel graph

Theorem 1. The graph obtained by duplication of each vertex of degree two in the gear graph is not a product cordial graph.

Proof. Let

Case 1: n is odd. Label the first

Hence, we get

Thus, it does not admit product cordial labeling.

Case 2: n is even. Label the first

Hence, we get

Select two vertices a and b from the sequence

Let A and B be the set of all vertices with label 0 and 1 respectively. If we interchange the labels of k many vertices from set A to k many vertices of set B then

So the graph G does not admit product cordial labeling. Hence, G is not a product cordial graph.

Theorem 2. The graph obtained by duplication of each vertex of degree two by an edge in the gear graph is a product cordial graph.

Proof. Let

Define a function

Thus

Illustration 1. The product cordial labeling of the graph obtained by duplication of each vertex of degree two by an edge in gear graph

Theorem 3. The graph obtained by duplication of each vertex of degree three by an edge in the gear graph is a product cordial graph.

Proof. Let

Define a function

Thus,

Illustration 2. The product cordial labeling of the graph obtained by duplication of each vertex of degree three by an edge in gear graph

Theorem 4. The graph obtained by duplication of the apex vertex in the gear graph by an edge admits product cordial labeling if n is even.

Proof. Let

Label the first

Illustration 3. The product cordial labeling of the graph obtained by duplication of the apex vertex by an edge in gear graph

Theorem 5. The graph obtained by switching of a vertex of degree two in gear graph is a product cordial graph.

Proof. Let

Case 1: n is odd. Label the first n vertices of the sequence

Case 2: n is even.

Subcase I:

Label first

label 0 and the remaining vertices each with label 1.

Subcase II:

Label first

with label 0 and the remaining vertices each with label 1.

From both the subcases we get

Clearly from both the cases

Illustration 4. The graph obtained by switching of a vertex of degree two in gear graph

Theorem 6. The graph obtained by applying vertex switching on a single vertex of degree three in gear graph is a product cordial graph.

Proof. Let

Case 1: n is odd. Label the first n vertices of the sequence

label 1. Thus

Case 2: n is even. Label the first n vertices of the sequence

label 1. By this labeling we get

Clearly from both the cases

Illustration 5. The graph obtained by switching of a vertex of degree three in gear graph

Observation 1. The graph obtained by applying vertex switching on the apex vertex in gear graph is a product cordial graph if n is odd and it is not product cordial graph if n is even.

The graph obtained by vertex switching of the apex vertex in the gear graph again yields us a gear graph. Vaidya and Barasara [

Conjecture 1. The graph obtained by duplication of each vertex of degree three in the gear graph is not a product cordial graph.

Conjecture 2. The graph obtained by duplication of the apex vertex in the gear graph is not a product cordial graph.

Conjecture 3. The graph obtained by switching of a vertex of degree two in gear graph is a product cordial graph.

We have derived seven results on product cordial labeling of some graphs obtained by duplication of some graph elements in gear graph. Also, we have derived two results on product cordial labeling of some graphs obtained by switching of a vertex of different degrees in the gear graph. Similar problems can be discussed for the graph obtained by duplication of an edge in gear graph. Also, the product cordial labeling can be discussed in the context of these graph operations of wheel graph, helm graph and crown graph.

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 Raval, K.K. (2016) Product Cordial Graph in the Context of Some Graph Operations on Gear Graph. Open Journal of Discrete Mathematics, 6, 259-267. http://dx.doi.org/10.4236/ojdm.2016.64022