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The k-rainbow domatic number of a graph

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Sheikholeslami S., Volkmann L.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 1

129-140

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2, ...,k} such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃_{u ∈ N(v)}f(u) = {1,2, ...,k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set ${f₁,f₂, ...,f_d}$ of k-rainbow dominating functions on G with the property that $∑_{i = 1}^d |f_i(v)| ≤ k$ for each v ∈ V(G), is called a k-rainbow dominating family (of functions) on G. The maximum number of functions in a k-rainbow dominating family on G is the k-rainbow domatic number of G, denoted by $d_{rk}(G)$. Note that $d_{r1}(G)$ is the classical domatic number d(G). In this paper we initiate the study of the k-rainbow domatic number in graphs and we present some bounds for $d_{rk}(G)$. Many of the known bounds of d(G) are immediate consequences of our results.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Sheikholeslami S.

1 Volkmann L.

1 2012

Wyniki wyszukiwania

The k-rainbow domatic number of a graph