Geodesic
On the dominator colorings in trees
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 677-683

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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 χ_d(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 ≤ χ_d(T) ≤ γ(T)+2. In this note we characterize nontrivial trees T attaining each bound.
Keywords: dominator coloring, domination, trees 
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Merouane, Houcine; Chellali, Mustapha. On the dominator colorings in trees. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 677-683. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a5/