Weak {2}-domination number of Cartesian products of cycles

For a graph (G=(V, E)), a weak ({2})-dominating function (f:Vrightarrow {0,1,2}) has the property that (sum _{uin N(v)}f(u)ge 2) for every vertex (vin V) with (f(v)= 0), where N(v) is the set of neighbors of v in G. The weight of a weak ({2})-dominating function f is the sum (sum _{vin V}f(v)) and the minimum weight of a weak ({2})-dominating function is the weak ({2})-domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak ({2})-domination number of (C_n Box C_5), which was obtained by St?pień, et al. (Discrete Appl Math 170:113–116, 2014). Moreover, we obtain the weak ({2})-domination numbers of (C_n Box C_3) and (C_n Box C_4).