Some results on cordiality labeling of generalized Jahangir graph

In this paper we consider the cordiality of a generalized Jahangir graph Jn,m. We give sufficient condition for Jn,m to admit (or not admit) the prime cordial labeling, product cordial labeling and total product cordial labeling.


Introduction
Let G = (V, E) be the connected, simple and undirected graph with vertex set V and edge set E(G).For standard terminology and notations in Graph Theory, we refer [18].By a labeling we mean any mapping that carries aset of graph elements to a set of numbers (usually positive integers), called labels.If the domain of the mapping is the set of vertices or the set of edges, then the labeling is called a vertex labeling or (edge labeling).If the domain is V ∪ E then we call the labeling as total labeling.Many labeling schemes have been introduced so far and they are well explored by many researchers.For a dynamic survey on various graph labeling problems, we refer to Gallian [3].
A labeling f : V (G) → {0, 1} is called binary vertex labeling of G and f (v) is called the label of the vertex v of G under f .If for an edge e = uv, the induced edge labeling is the number of vertices of G having label i under f and e f (i) is the number of edges of G having label i under f , where i = 0 or 1.
The concept of cordial labeling was introduced by Cahit [2] as a weaker version of graceful labeling and harmonious labeling.
The notion of prime labeling was originated by Entringer and was introduced by Tout et al. [17].Motivated by the concepts of prime labeling and cordial labeling, a new concept termed as a prime cordial labeling was introduced by Sundaram et al. [8] as follows.
The graph admits a prime cordial labeling is called a prime cordial graph.
Many graph families proved to be prime cordial, for example see [8,12,13,14].In 2004, Sundaram et al. [9] introduced the product cordial labeling of graph.Definition 3. Let f : V (G) → {0, 1} be a vertex labeling of a graph G that induces an edge labeling function In 2006, Sundaram et al. [10] introduced the notion of total product cordial labeling of graph.
For more results on product cordial and total product cordial, please refer [4,5,6,9,10,11].For n, m ≥ 2, the generalized Jahangir graph J n,m is a graph on nm + 1 vertices, that is, the graph consists of a cycle C mn with one additional vertex which adjacent to a m vertices of C mn at distance n to each other on C mn , see [1,7].The following figure shows the graph J n,m for n = 3 and m = 10.
In this paper, we investigate prime cordial labeling, product cordial labeling and total product cordial labeling of generalized Jahangir graph J n,m .

Main Results
In this section, we present our main results.Proof.Let J 2,m , m ≥ 4 be Jahangir graph with the vertex set V (J 2,m ) = {v} ∪ {v i : 1 ≤ i ≤ 2m} and the edge set To show that J 2,m is a prime cordial, we define a vertex labelingf : Hence, the Jahangir graph J 2,m is prime cordial.
This completes the proof.
Proof.Let J n,m , n > 2, m > 3 be Jahangir graph with the vertex set V (J n,m ) = {v}∪{v i : 1 ≤ i ≤ mn} and the edge set To show that J n,m is prime cordial, we define vertex labeling f : V (J n,m ) → {1, 2, . . ., mn+1} as follows: We have , if m are even and n are odd , for otherwise , if m are even and n are odd , for otherwise It is easy to show that |e f (0) − e f (1)| ≤ 1.Hence, the Jahangir graph J n,m is prime cordial.This completes the proof.
The following figure illustrates the prime cordial labeling of graph J 3,5 .
Theorem 2.2.The Jahangir graph J n,m is product cordial with n ≥ 2, m ≥ 3, m is odd and n is even.
Proof.Let J n,m with n is even and n ≥ 2, m is odd and m ≥ 3, be Jahangir graph with the vertex set V (J n,m ) = {v} ∪ {v i : 1 ≤ i ≤ mn} and the edge set www.ijc.or.idTo show that J n,m is product cordial, define a vertex labeling f : V (J n,m ) → {0, 1} in the following way: From the above labeling, we can see that This completes the proof.Figure 3 below illustrates the product cordial labeling of graph J 2,5 .In Theorem 2.2, the graph J n,m is product cordial labeling for n ≥ 2, m ≥ 3, m is odd and n is even.We have tried to find the product cordial labeling of J n,m for all values of m and n but so far without success.So we pose the following open problem.Problem 1. Determine product cordial labeling of the Jahangir graph J n,m for all m and n.
Proof.Let J n,m , n ≥ 2, m ≥ 3 be a Jahangir graph with the vertex set V (J n,m ) = {v} ∪ {v i : 1 ≤ i ≤ mn} and the edge set To show that J n,m is total product cordial, define a vertex labeling f : V (J n,m ) → {0, 1} in the following way: Case 1: m and n are odd.
Case 2: m and n are not odd.This completes the proof.Figure 4 shows the total product cordial labeling of graph J 4,5 .In [15,16], Vaidya and Barasara introduced an edge product cordial labeling and a total edge product cordial labeling of graph G. Thus, we propose the following problem.Problem 2. Determine edge product cordial labeling and total edge product cordial labeling of the Jahangir graph J n,m for n, m ≥ 2.

Figure 1 .
Figure 1.Jahangir graph J 3,10 for m, n are odd 9, for otherwise f (v mn−2 ) = m, n are even or m, n are odd 5, for otherwise Some results on cordiality labeling ... | R. Hasni, S. Matarneh and A. Azaizeh