Geodesic
The k-rainbow domatic number of a graph
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 129-140.

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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.
Keywords: k-rainbow dominating function, k-rainbow domination number, k-rainbow domatic number 
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Sheikholeslami, Seyyed; Volkmann, Lutz. The k-rainbow domatic number of a graph. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 129-140. http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a10/

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