Total dominator coloring number of middle 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 of a graph is the minimum number of color classes in a total dominator coloring. In this paper, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitly the total dominator chromatic number of the middle graph of several known families of graphs.