Some new families of 3-equitable prime cordial graphs

and the label 0 otherwise, then the number of edges labeled with i and the number of edges labeled with j differ by at most 1 for . 2 0 2 0     j and i

If a graph has a 3-equitable prime cordial labeling, then it is called a 3equitable prime cordial graph. In this paper, we have prove that the bistar ) 2 ( ,

Introduction

Graph labeling is an active area of research in graph theory which has rigorous applications in coding theory, communication networks, optimal circuits layouts and crystallography. In this paper, we consider only finite, simple undirected graphs. We consider a graph G = (V (G), E (G)) and we let |V (G)| = p and |E (G)| = q. For graph theoretic notations and terminology we follow Harary [4] and for number theory we follow Burton [1] . A labeling of a graph G is a mapping that carries vertices and/or edges into a set of numbers, usually integers. An excellent survey of graph labeling and various types of graph labeling can be found in Gallian [3] . We provide the brief summary of results which will be useful for the present investigations.

Definition 1.1: Let G be a graph. A mapping

For an edge uv e = , the induced edge labeling

We denote vf (i) is the number of vertices of G having label i and ef (i) is the number of edges of G having label i under f * .

Definition 1.2: A binary vertex labeling f of a graph G is called a cordial labeling if

The concept of cordial labeling was introduced by Cahit [2] .

Definition 1.3: A prime cordial labeling of a graph G with vertex set

A graph which admits prime cordial labeling is called a prime cordial graph. The concept of prime cordial labeling was introduced by Sundaram et al [6] . Definition 1.4: A 3-equitabile prime cordial labeling of a graph G with vertex set

A graph which admits 3-equitable prime cordial labeling is called a 3-equitable prime cordial graph. The concept of 3-equitable prime cordial labeling was introduced by Murugesan et al. [5,7] , the following graphs are proved to have 3-equitable prime cordial labeling: path Pn, cycle Cn (n ≥ 4), star graph K(1, n) if n ≡ 1(mod 3), complete graph Kn (n ≤ 2), cycle with one chord, cycle with twin cord split graph. In this paper, we prove that the following graphs are 3-equitable prime cordial graphs. Theorem 1.9 The bistar Bn,n is a 3-equitable prime cordial graph for n ≥ 2.

Proof: Let u1, u2,..., un and v1, v2,..., vn be the pendent vertices with u, v are the apex vertices of Bn,n. Let G be the graph Bn,n.

From the definition of f, we have

Hence Bn,n is 3-equitable prime cordial graph for n ≥ 2.

Example 1.10 The 3-equitable prime cordial labeling of graphs B(4, 4) and B(9, 9) are shown in Figure 1 and Figure 2 respectively. To define f :

→ {1, 2, 3..., 2n}, we consider two cases.

Case 1: when n is an even integer. We define Further, we define f (un) = 2n -1 and f (vn) = 2n.

Sub case 2.2. n ≡ 1(mod3)

We define f (ui) and f (vi) for 1 ≤ i ≤ n -3 as same as The vertex labeling of P2 × Pn for 3 ≤ n ≤ 4 is shown in the Table 1.

We define the vertex labeling f: V (G) → {1, 2,... 2n} as follows.

f (v4) = 7 f (ui) = 2i for i = 5, 6,..., n. f (vj) = 2j -1 for j = 5, 6,..., n. Then

Hence SLn is a 3-equitable prime cordial graph.

Remark 1.16

The ladder P2 is not a 3-equitable prime cordial labeling.

Example 1.14 The 3-equitable prime cordial labeling of graph SL7 is shown in Figure 6. 2 for n = 1, 2, 3.

We define the vertex labeling of K(3, n) for n > 3 is divided into three cases.

Concluding Remarks

In the present paper we determine the 3-equitable prime cordial labeling of the graphs such as bistar, comb, ladder, kite and slanting ladder. It is an interesting open area of research to find out the graphs which admit 3-equitable prime cordial labeling.

References

1. David M Burton. Elementary Number theory, Second Edition, Wm. C. Brown Company Publishers, 1980.
2. Cahit I. On Cordial and 3-equitable Labelings of Graphs, Utilitas Math, 1990; 37:189-198.
3. Gallian JA. A Dyanamic Survey of Graph Labeling, Electronic Journal of Combinatorics, 2016; 16(6).
4. Harary F. Graph Theory, Addition-Wesley, Reading Mass, 1972.
5. Murugesan S. 3-equitable Prime cordial Labeling of Graphs, International Journal of Applied Information Systems. 2013; 5:1-4.
6. Sundaram M, Ponraj R, Somasundaram S. Prime cordial labeling of Graphs, Journal of Indian Academy of Mathematics, 2005; 27:373-390.
7. Dr. Sweta Srivastav, Dr. Sangeeta Gupta. 3-equitable Prime cordial Labeling of Some Graphs, International Journal of Engineering Research. 2015; 4(3):115-117.