Nordhaus-Gaddum bounds on the k-rainbow domatic number of a graph

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, . . . , fd} of k-rainbow dominating functions on G with the property that ∑ d i=1 |fi(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 krainbow dominating family on G is the k-rainbow domatic number of G, denoted by drk(G). Note that dr1(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 drk(G). Many of the known bounds of d(G) are immediate consequences of our results.