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On the dominator colorings in trees

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Merouane H., Chellali M.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 4

677-683

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.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Chellali M.

1 Merouane H.

1 2012

Wyniki wyszukiwania

On the dominator colorings in trees