The k-Rainbow Domination and Domatic Numbers of Digraphs

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v is an element of V (D) with f(v) = (sic), the condition boolean OR(u is an element of N-(v)) f(u) = {1, 2, . . . , k} is fulfilled, where N (v) is the set of in-neighbors of v. A set {f(1), f(2), . . . , f(d)} of k-rainbow dominating functions on D with the property that Sigma(d)(i=1) |f(i) (v)| <= k for each v is an element of V (D), is called a k-rainbow dominating family (of functions) on D. The maximum number of functions in a k-rainbow dominating family on D is the k-rainbow domatic number of D, denoted by d(rk)(D). In this paper we initiate the study of the k-rainbow domatic number in digraphs, and we present some bounds for d(rk)(D).