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Discussiones Mathematicae Graph Theory 32(4) (2012)
677-683
DOI: https://doi.org/10.7151/dmgt.1635
On the dominator colorings in trees
| Houcine Boumediene Merouane and Mustapha Chellali
LAMDA-RO, Department of Mathematics |
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
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