The k-Rainbow Domination and Domatic Numbers of Di- graphs
نویسندگان
چکیده
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 u∈N−(v) f(u) = {1, 2, . . . , k} is fulfilled, 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 ∑d 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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