Total Dominator Colorings and Total Domination in Graphs

A total dominator coloring of a graph $$G$$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 (Formula presented.) 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 (Formula presented.) 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 (Formula presented.), where (Formula presented.) denotes the chromatic number of G. We establish properties of total dominator colorings in trees. We characterize the trees T for which (Formula presented.). We prove that if $$T$$T is a tree of $$n \ge 2$$n≥2 vertices, then (Formula presented.) and we characterize the trees achieving equality in this bound.