On the dominator colorings in trees

Abstract

In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different colors. A dominator coloring of a graph G is a proper coloring such that every vertex of V dominates all vertices of at least one color class (possibly its own class). The dominator chromatic number χd(G) is the minimum number of color classes in a dominator coloring of G. Gera showed that every nontrivial tree T satisfies γ(T) + 1 ≤ χd(T) ≤ γ(T) + 2. In this note we characterize nontrivial trees T attaining each bound.