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

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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

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.

Keywords: dominator coloring, domination, trees

2010 Mathematics Subject Classification: 05C69, 05C15.

References

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