Dominator and total dominator chromatic number of Mongolian tent and fire cracker graphs

Abstract

A dominator coloring of a graph $G$ is a proper coloring in which every vertex of $G$ dominates at least one color class. The dominator chromatic number of $G, \chi_d(G)$, is defined by the minimum number of colors needed in a dominator coloring of $G$. A total dominator coloring of $G$ is a proper coloring of $G$ with the extra property that every vertex in $G$ properly dominates a color class. The total dominator chromatic number of $G, \chi_{t d}(G)$, is the minimum number of colors needed in a total dominator coloring of $G$. In this paper, we obtain dominator and total dominator chromatic number of Mongolian tent graphs and fire cracker graphs.