Total dominator total chromatic numbers of cycles and paths

Abstract

The total dominator total coloring of a graph is a total coloring of the graph such that each object (vertex or edge) of the graph is adjacent or incident to every object of some color class. The minimum number of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. In (A.P. Kazemi, F. Kazemnejad and S. Moradi, Contrib. Discrete Math. (2022).), the authors initiated to study the total dominator total coloring of a graph and found some useful results, and presented some problems. Finding the total dominator total chromatic numbers of cycles and paths were two of them which we consider them here.