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For a graph $$G=(V, E)$$ , a weak $$\{2\}$$ -dominating function $$f:V\rightarrow \{0,1,2\}$$ has the property that $$\sum _{u\in N(v)}f(u)\ge 2$$ for every vertex $$v\in V$$ 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 $$\sum _{v\in V}f(v)$$ 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 Stȩpień, et al. (Discrete Appl Math 170:113-116, 2014). Moreover, we obtain the weak $$\{2\}$$ -domination numbers of $$C_n \Box C_3$$ and $$C_n \Box C_4$$ . [ABSTRACT FROM AUTHOR]

Let $$G=(V, E)$$ be a graph. Denote $$d_G(u, v)$$ the distance between two vertices u and v in G. An L(2, 1)-labeling of G is a function $$f: V \rightarrow \{0,1,\ldots \}$$ such that for any two vertices u and v, $$|f(u)-f(v)| \ge 2$$ if $$d_G(u, v) = 1$$ and $$|f(u)-f(v)| \ge 1$$ if $$d_G(u, v) = 2$$ . The span of f is the difference between the largest and the smallest number in f( V). The $$\lambda $$ -number $$\lambda (G)$$ of G is the minimum span over all L(2, 1)-labelings of G. In this paper, we conclude that the $$\lambda $$ -number of each brick product graph is 5 or 6, which confirms Conjecture 6.1 stated in Li et al. (J Comb Optim 25:716-736, 2013). [ABSTRACT FROM AUTHOR]

Let $$G=(V, E)$$ be a graph. Denote $$d_G(u, v)$$ the distance between two vertices $$u$$ and $$v$$ in $$G$$ . An $$L(2, 1)$$ -labeling of $$G$$ is a function $$f: V \rightarrow \{0,1,\cdots \}$$ such that for any two vertices $$u$$ and $$v$$ , $$|f(u)-f(v)| \ge 2$$ if $$d_G(u, v) = 1$$ and $$|f(u)-f(v)| \ge 1$$ if $$d_G(u, v) = 2$$ . The span of $$f$$ is the difference between the largest and the smallest number in $$f(V)$$ . The $$\lambda $$ -number of $$G$$ , denoted $$\lambda (G)$$ , is the minimum span over all $$L(2,1 )$$ -labelings of $$G$$ . In this article, we confirm Conjecture 6.1 stated in X. Li et al. (J Comb Optim 25:716-736, ) in the case when (i) $$\ell $$ is even, or (ii) $$\ell \ge 5$$ is odd and $$0 \le r \le 8$$ . [ABSTRACT FROM AUTHOR]