ISSN (0970-2083)

NEW CLASSES OF K4 SNAKE GLUING OF CORDIAL GRAPHS

Tamilselvi L1*, 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

Visit for more related articles at Journal of Industrial Pollution Control

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 mappingequation 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 labelingequation is given byequation 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 equation

The concept of cordial labeling was introduced by (CahitI, 1987). In the same paper author proved that tree is cordial and Kn 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 equation 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 K4SN in one copy. The graph G2 = Cm (K4Sn)t is defined similarly (Ho, et alet al., 1989; Sundaram, et al., 2006; Tamilselvi, 2013) (Figure. 1).

icontrolpollution-Snake

Figure 1: K4 Snake graph K4SN.

Theorem-1

The Graph equation is cordial.

Proof

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

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

icontrolpollution-Path-graph

Figure 2: Path graph G1 = PM(K4Sn)t.

equation

equation

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

equation

The edge set is defined as

equation

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

equation

Theorem-2

The graph

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

icontrolpollution-Cordial-labeling

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

icontrolpollution-labeling

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

  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
vf (0) = vf (1) ef (0) = ef (1) + 1
Table 1. Showing vertex conditions and edge conditions of equation

Proof

The graph equation 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:

equation

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

icontrolpollution-Cycle

Figure 5: Cycle graph Cm(K4Sn)t.

icontrolpollution-Cordial

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
equation equation

Table 2. Showing vertex conditions and edge conditions of equation

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

Cahit, I. (1987). Cordial graphs, A weaker version of graceful and harmonious Graphs. ArsCombinatoria. 23 : 201-207.

Gallian, J.A. (2011). A dynamic survey of graph labeling. the electronic J. Combinatorics. 5 : DS6. http://www.combinatorics.org.

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.

Sundaram, M., Ponraj, R. and Somasundaram, S. (2006). Total Product Cordial labeling of Graphs. Bull Pure and applied sciences (Mathematics and statistics). 199-203.

Tamilselvi, L. (2013). New classes of graphs relating to quadrilateral snake using valuation, odd graceful, felicitous. Mean and Cordial labeling.
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