Total Dominator Colorings and Total Domination in Graphs

Abstract

A total dominator coloring of a graph $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi _d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. A total dominating set of $G$ is a set $S$ of vertices such that every vertex in $G$ is adjacent to at least one vertex in $S$. The total domination number $\gamma _t(G)$ of $G$ is the minimum cardinality of a total dominating set of $G$. We establish lower and upper bounds on the total dominator chromatic number of a graph in terms of its total domination number. In particular, we show that every graph $G$ with no isolated vertex satisfies $\gamma _t(G) \le \chi _d^t(G) \le \gamma _t(G) + \chi (G)$, where $\chi (G)$ denotes the chromatic number of $G$. We establish properties of total dominator colorings in trees. We characterize the trees $T$ for which $\gamma _t(T) = \chi _d^t(T)$. We prove that if $T$ is a tree of $n \ge 2$ vertices, then $\chi _d^t(T) \le 2(n+1)/3$ and we characterize the trees achieving equality in this bound.