ISSN (0970-2083)

All submissions of the EM system will be redirected to **Online Manuscript Submission System**. Authors are requested to submit articles directly to **Online Manuscript Submission System** of respective journal.

**Tamilselvi L ^{1*}, Gayathri S^{1}, Selvaraju P^{2}, Vemuri Lakshminarayana^{3}**

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

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

^{3}Principal, 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

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

Graph labeling, Cordial labeling, Cycle graph, Path graph, K_{4} Snake graph

**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

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

The graph is defined as an isomorphic *K _{4}* snake‘t’ copies gluing with each

**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**).

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**).

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 |
v (0) = _{f}v (1)_{f} |
e_{f }(0) = e_{f} (1) + 1 |

It is clear thatis cordial.

**Proof**

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

Vertex labeling of G_{2} is same as in theorem-1. The
edge set is defined as:

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

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 |

It is clear that is cordial.

**Table 2.** Showing vertex conditions and edge conditions of

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 *K _{4}* snake‘t’ copies gluing
of path graph and Cycle graph.

Gallian, J.A. (2011). A dynamic survey of graph labeling.

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.

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

Copyright © 2019 Research and Reviews, All Rights Reserved