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/

[1] M. Chellali and F. Maffray, Dominator colorings in some classes of graphs, Graphs Combin. 28 (2012) 97-107, doi: 10.1007/s00373-010-1012-z.

[2] R. Gera, On the dominator colorings in bipartite graphs in: Proceedings of the 4th International Conference on Information Technology: New Generations (2007) 947-952, doi: 10.1109/ITNG.2007.142.

[3] R. Gera, On dominator colorings in graphs, Graph Theory Notes of New York LII (2007) 25-30.

[4] R. Gera, S. Horton and C. Rasmussen, Dominator colorings and safe clique partitions, Congr. Numer. 181 (2006) 19-32.

[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).

[6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc., New York, 1998).

[7] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, 1962).

[8] L. Volkmann, On graphs with equal domination and covering numbers, Discrete Appl. Math. 51 (1994) 211-217, doi: 10.1016/0166-218X(94)90110-4.