ISSN (0970-2083)

**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

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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.

**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 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.

- 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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