Journal of Discrete Mathematics

Volume 2015, Article ID 512696, 6 pages

http://dx.doi.org/10.1155/2015/512696

## Product Cordial and Total Product Cordial Labelings of

^{1}College of Science, Harbin Engineering University, Harbin 150001, China^{2}Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Segamat Campus, 85000 Johor, Malaysia

Received 23 July 2014; Revised 30 September 2014; Accepted 15 December 2014

Academic Editor: Kinkar Ch. Das

Copyright © 2015 Zhen-Bin Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We proved that is total product cordial. We also give sufficient conditions for the graph to admit (or not admit) a product cordial labeling.

#### 1. Introduction

Let denote a simple and finite connected graph with vertex set and edge set . Suppose and denote a vertex and an edge labeling of a graph, respectively. Let and denote the number of vertices and edges labeled with . In 2004, Sundaram et al. [1] introduced the notion of product cordial labelings.

*Definition 1. *Let be a vertex labeling of a graph that induces an edge labelings such that . One says is a product cordial labeling if and . A graph is called a product cordial graph if it admits a product cordial labeling.

Sundaram et al. [1] proved that many graphs are product cordial: trees; unicyclic graphs of odd order; triangular snakes; dragons; helms; ; if and only if is odd; ; ; ; ; ; ; if and only if is odd; if and only if . Kwong et al. [2] discussed product cordial index sets of 2 regular graphs. Kwong et al. [3] discussed product cordial index sets of cylinders.

In 2006, Sundaram et al. [4] introduced the notion of total product cordial labelings.

*Definition 2. *Let be a vertex labeling of a graph that induces edge labelings such that . We say is a total product cordial labeling if . A graph is called a total product cordial graph if it admits a total product cordial labeling.

Sundaram et al. [4, 5] also proved that graphs are total product cordial: every product cordial graph of even order or odd order and even size; trees; all cycles except ; ; with edges appended at each vertex; fans; wheels; helms. In [6], Ramanjaneyulu et al. proved that a family of planar graphs for which each face is a 4-cycle admit a total product cordial labeling.

In this paper, we determine the product cordiality and total product cordiality of the th power of the path , denoted by , which is defined as follows [7].

*Definition 3. *Let denote a path of length . The graph is obtained from by adding edges that join all vertices and whose distance is . The graph is illustrated in Figure 1. has edges.