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Abstract
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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 ∈ V (D) with f(v) = ∅ the condition ∪u2N(v) f(u) = {1; 2; : : : ; k}
is ful lled, where N(v) is the set of in-neighbors of v. A set {f1; f2; : : : ; fd} of k-rainbow
dominating functions on D with the property that Σd
i=1 |fi(v)| ≤ k for each v ∈ 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 drk(D). In this paper we initiate the study of the k-rainbow domatic number
in digraphs, and we present some bounds for drk(D).
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