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

For a graph $$G=(V, E)$$ G = ( V , E ) , a weak $$\{2\}$$ { 2 } -dominating function $$f:V\rightarrow \{0,1,2\}$$ f : V → { 0 , 1 , 2 } has the property that $$\sum _{u\in N(v)}f(u)\ge 2$$ ∑ u ∈ N ( v ) f ( u ) ≥ 2 for every vertex $$v\in V$$ v ∈ V with $$f(v)= 0$$ f ( v ) = 0 , where N(v) is the set of neighbors of v in G. The weight of a weak $$\{2\}$$ { 2 } -dominating function f is the sum $$\sum _{v\in V}f(v)$$ ∑ v ∈ V f ( v ) and the minimum weight of a weak $$\{2\}$$ { 2 } -dominating function is the weak $$\{2\}$$ { 2 } -domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak $$\{2\}$$ { 2 } -domination number of $$C_n \Box C_5$$ C n □ C 5 , which was obtained by Stȩpień, et al. (Discrete Appl Math 170:113–116, 2014). Moreover, we obtain the weak $$\{2\}$$ { 2 } -domination numbers of $$C_n \Box C_3$$ C n □ C 3 and $$C_n \Box C_4$$ C n □ C 4 .