Total dominator total chromatic numbers of cycles and paths

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