LAUSR

A k-rainbow dominating function (kRDF) of G is a function $$f:V(G)\rightarrow {\mathcal {P}}(\{1,2,\ldots ,k\})$$f:V(G)→P({1,2,…,k}) for which $$f(v)=\emptyset $$f(v)=∅ we have $$\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2,\ldots ,k\}$$⋃u∈N(v)f(u)={1,2,…,k}. The weightw(f) of a function f is defined as $$w(f)=\sum _{v\in V(G)}\left| f(v)\right| $$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 $$\gamma _{rk}(G)$$γrk(G). In this paper, we determine the exact values of the 2-rainbow domination numbers of $$C_4\Box C_n$$C4□Cn and $$C_8\Box C_n$$C8□Cn. It follows that $$\gamma _{r2}\not = 2\gamma $$γr2≠2γ for graphs $$C_4\Box C_n$$C4□Cn ($$n \ge 4$$n≥4) and $$C_8\Box C_n$$C8□Cn ($$n \ge 8$$n≥8), answering in part a question raised by Brešar.