2-Rainbow domination number of Cartesian products: C_{n}\square C_{3} and C_{n}\square C_{5}

A function \(f:V(G)\rightarrow \mathcal P (\{1,\ldots ,k\})\) is called a \(k\) -rainbow dominating function of \(G\) (for short \(kRDF\) of \(G)\) if \( \bigcup \nolimits _{u\in N(v)}f(u)=\{1,\ldots ,k\},\) for each vertex \( v\in V(G)\) with \(f(v)=\varnothing .\) By \(w(f)\) we mean \(\sum _{v\in V(G)}\left|f(v)\right|\) and we call it the weight of \(f\) in \(G.\) The minimum weight of a \( kRDF\) of \(G\) is called the \(k\) -rainbow domination number of \(G\) and it is denoted by \(\gamma _{rk}(G).\) We investigate the \(2\) -rainbow domination number of Cartesian products of cycles. We give the exact value of the \(2\) -rainbow domination number of \(C_{n}\square C_{3}\) and we give the estimation of this number with respect to \(C_{n}\square C_{5},\) \((n\ge 3).\) Additionally, for \(n=3,4,5,6,\) we show that \(\gamma _{r2}(C_{n}\square C_{5})=2n.\)