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 _{uin N(v)}f(u)={1,ldots ,k},) for each vertex ( vin V(G)) with (f(v)=varnothing .) By (w(f)) we mean (sum _{vin 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},) ((nge 3).) Additionally, for (n=3,4,5,6,) we show that (gamma _{r2}(C_{n}square C_{5})=2n.)