The 2-rainbow domination number of Cartesian product of cycles

A k-rainbow dominating function (kRDF) of G is a function that assigns subsets of {1, 2, ..., k} to the vertices of G such that for vertices v with f(v) = ∅ we have ⋃u ∈ N(v)f(u) = {1, 2, ..., k}. The weight w(f) of a kRDF f is defined as w(f) = ∑v ∈ V(G)|f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, which is denoted by γrk(G). In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases.