Total dominator chromatic number of $P_m \times C_n$

Abstract

A total dominator coloring of a graph $G$ with $\delta(G) \geq 1$ is a proper coloring of $G$ with the extra property that every vertex in $G$ properly dominates a color class. The total dominator chromatic number of $G$ is the minimum number of colors needed in a total dominator coloring of $G$, denoted by $\chi_{t d}(G)$. In this paper, we obtain total dominator chromatic number of $P_m \times C_n$.