Total dominator coloring of the lexicographic product of graphs

A total dominator coloring of a graph which has no isolated vertex is a proper vertex coloring of the graph in which each vertex of the graph is adjacent to all vertices of some (other) color class. The total dominator chromatic number of a graph is the minimum size of color classes in a total dominator coloring of the graph. This paper establishes the total dominator coloring of the lexicographic product of graphs. Firstly, we present some graphs whose total dominator chromatic numbers are same. Also we show some graphs whose total dominator chromatic numbers are equal to their chromatic numbers. Then, after presenting four upper bounds and a lower bound for the total dominator chromatic number of the lexicographic product of two graphs, we give a sufficient condition for that the total dominator chromatic number of the lexicographic product of two graphs to be equal to the total dominator chromatic number of the lexicographic product of the first graph to a complete graph. We next establish a Nordhaus-Gaddum-like relation for the total dominator chromatic number of the lexicographic product of two graphs, and finally find the total dominator chromatic number of the lexicographic product of a star, a wheel or a complete graph with an arbitrary graph.