Total dominator coloring in product graphs

Abstract

A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number χ^t_d(G) of G is the minimum number of color classes in a total dominator coloring of it. In [Total dominator chromatic number in graphs, submitted] the author initiated to study this number in graphs and continued his studying in [Total dominator chromatic number and Mycieleskian graphs, submitted]. In this paper, we study total dominator colorings in the Cartesian product and the cross product of graphs.