The 2-Rainbow Domination Numbers of $${\boldsymbol{C}}_4\Box {\boldsymbol{C}}_n$$ C 4 □ C n and $${\boldsymbol{C}}_8\Box {\boldsymbol{C}}_n$$ C 8 □ C n

Abstract

A k-rainbow dominating function (kRDF) of G is a function $$f:V(G)\rightarrow {\mathcal {P}}(\{1,2,\ldots ,k\})$$ for which $$f(v)=\emptyset $$ we have $$\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2,\ldots ,k\}$$ . The weightw(f) of a function f is defined as $$w(f)=\sum _{v\in V(G)}\left| f(v)\right| $$ . The minimum weight of a kRDF of G is called the k-rainbow domination number of G, which is denoted by $$\gamma _{rk}(G)$$ . In this paper, we determine the exact values of the 2-rainbow domination numbers of $$C_4\Box C_n$$ and $$C_8\Box C_n$$ . It follows that $$\gamma _{r2}\not = 2\gamma $$ for graphs $$C_4\Box C_n$$ ( $$n \ge 4$$ ) and $$C_8\Box C_n$$ ( $$n \ge 8$$ ), answering in part a question raised by Bresar.