Discussiones Mathematicae Graph Theory 32(4) (2012) 677-683
doi: 10.7151/dmgt.1635

[BIBTex] [PDF] [PS]

On the dominator colorings in trees

Houcine Boumediene Merouane and Mustapha Chellali

LAMDA-RO, Department of Mathematics
University of Blida
B. P. 270, Blida, Algeria


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.

Keywords: dominator coloring, domination, trees

2010 Mathematics Subject Classification: 05C69, 05C15.


[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.

Received 24 January 2011
Revised 11 August 2011
Accepted 19 December 2011