NEW CLASSES OF K4 SNAKE GLUING OF CORDIAL GRAPHS

Tamilselvi L^1*, Gayathri S¹, Selvaraju P², Vemuri Lakshminarayana³

¹Department of Mathematics, Aarupadai Veedu Institute of Technology (AVIT), Vinayaka Missions University, Chennai, 603104, India

²Department of Mathematics, Vel Tech Multitech, Dr. Rangarajan and Dr. Sakunthala Engineering College, India

³Principal, Vinayaka Missions University, Chennai, 603104, India

*Corresponding Author:

Tamilselvi L
Department of Mathematics
Aarupadai Veedu Institute of Technology
(AVIT), Vinayaka Missions University
Chennai, 603104, India
E-mail: ltamilselvi@avit.ac.in

Received Date: 17 June, 2017 Accepted Date: 22 August, 2017

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Abstract

The graph K4 SN is called a K4 K4 Snake graph. The vertex set V and edge set E are described below

Keywords

Graph labeling, Cordial labeling, Cycle graph, Path graph, K₄ Snake graph

Introduction

Definition 1.1

Let G = (V(G), E(G)) be a graph. A mapping is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. For an edge e = uv, the induced edge labeling is given by Let (0), (1) f f v v be the number of vertices of G having labels 0 and 1 respectively under f and let ef (0), ef (1) be the number of edges having labels 0 and 1 respectively under f *.

Definition 1.2

A binary vertex labeling of a graph G is called a cordial labeling if

The concept of cordial labeling was introduced by (CahitI, 1987). In the same paper author proved that tree is cordial and K_n is cordial if and only if n ≤ 3.

For an exhaustive survey of these topics one may refer to the excellent survey paper of (Gallian, 2011).

Main Results

The graph is defined as an isomorphic K₄ snake‘t’ copies gluing with each m. N is the number of Blocks (i.e., K₄) of K₄ snake K₄S_N in one copy. The graph G₂ = C_m (K₄S_n)t is defined similarly (Ho, et alet al., 1989; Sundaram, et al., 2006; Tamilselvi, 2013) (Figure. 1).

Figure 1: K₄ Snake graph K₄S_N.

Theorem-1

The Graph is cordial.

Proof

The graph has m(1 + 3tn) vertices and m(6t n +1) −1 edges.

We define vertex labeling of as follows (Figure. 2).

Figure 2: Path graph G₁ = P_M(K₄S_n)^t.

For copies are incident with ‘0’ and ‘1’ are

The edge set is defined as

Copies are incident with the vertices assigned the label ‘0’ and ‘1’

Theorem-2

The graph

is Cordial (Table 1) and (Figures. 3 and 4).

Figure 3: Cordial labeling of P₄(K₄S₃)².

Figure 4: Cordial labeling of P₈(K₄S₅)¹.

Table 1. Showing vertex conditions and edge conditions of

Proof

The graph has m(1+ 3tn) vertices and m(1+ 6t n) edges.

Vertex labeling of G₂ is same as in theorem-1. The edge set is defined as:

The remaining edge labeling of G₂ is same as in theorem-1 (Figures. 5 and 6) (Table 2).

Figure 5: Cycle graph C_m(K₄S_n)^t.

Figure 6: Cordial labeling of C₄(K₄S₃)³.

Number of copies (t)	Number of block  (n)	Vertex conditions	Edge conditions
t is even (or) t is odd	n is even (or) n is odd

Table 2. Showing vertex conditions and edge conditions of

Conclusion

According to literature survey, more work has been done in cordial labeling for cycle and path related graphs. In our work we determine the cordial labeling for new classes of K₄ snake‘t’ copies gluing of path graph and Cycle graph.

References

1. Cahit, I. (1987). Cordial graphs, A weaker version of graceful and harmonious Graphs. ArsCombinatoria. 23 : 201-207.
2. Gallian, J.A. (2011). A dynamic survey of graph labeling. the electronic J. Combinatorics. 5 : DS6. http://www.combinatorics.org.
3. Ho, Y.S., Lee, S.M. and Shee, S.C. (1989). Cordial labeling of unicycle graphs and generalized Petersen graphs. Congress. Number. 68 : 109-122.
4. Sundaram, M., Ponraj, R. and Somasundaram, S. (2006). Total Product Cordial labeling of Graphs. Bull Pure and applied sciences (Mathematics and statistics). 199-203.
5. Tamilselvi, L. (2013). New classes of graphs relating to quadrilateral snake using valuation, odd graceful, felicitous. Mean and Cordial labeling.