For a positive integer k, a k-rainbow dominating function (kRDF) 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 with f(v) = ∅, u∈N(v), f(u) = {1, 2, . . . , k}, where N^{-}(v) is the set of in-neighbors of v. The weight of a kRDF f of D is the value v∈V (D), |f(v)|. The k-rainbow domination number of a digraph D, denoted by γrk(D), is the minimum weight of a kRDF of D. Let PmPn denote the Cartesian product of Pm and P_n, where Pm and Pn denote the directed paths of order m and n, respectively. In this paper, we determine the exact values of γrk(PmPn) for any positive integers k ≥ 2, m and n.
