On the total product cordial labeling on the cartesian product of $P_m \times C_n$, $C_m \times C_n$ and the generalized Petersen graph $P(m, n)$

Abstract

A total product cordial labeling of a graph $G$ is a function $f: V \rightarrow\{0,1\}$. For each $x y$, assign the label $f(x) f(y), f$ is called total product cordial labeling of $G$ if it satisfies the condition that $\mid v_f(0)+e_f(0)-$ $v_f(1)-e_f(1) \mid \leq 1$ where $v_f(i)$ and $e_f(i)$ denote the set of vertices and edges which are labeled with $i=0,1$, respectively. A graph with a total product cordial labeling defined on it is called total product cordial.
 In this paper, we determined the total product cordial labeling of the cartesian product of $P_m \times C_n, C_m \times C_n$ and the generalized Petersen graph $P(m, n)$.