Total dominator chromatic number of a graph

Abstract

‎Given a graph $G$‎, ‎the total dominator coloring problem seeks a‎ ‎proper coloring of $G$ with the additional property that every‎ ‎vertex in the graph is adjacent to all vertices of a color class‎. ‎We‎ ‎seek to minimize the number of color classes‎. ‎We initiate to study‎ ‎this problem on several classes of graphs‎, ‎as well as finding‎ ‎general bounds and characterizations‎. ‎We also compare the total‎ ‎dominator chromatic number of a graph with the chromatic number and‎ ‎the total domination number of it‎.