LAUSR

Let $$G=(V, E)$$G=(V,E) be a graph. Denote $$d_G(u, v)$$dG(u,v) the distance between two vertices u and v in G. An L(2, 1)-labeling of G is a function $$f: V \rightarrow \{0,1,\ldots \}$$f:V→{0,1,…} such that for any two vertices u and v, $$|f(u)-f(v)| \ge 2$$|f(u)-f(v)|≥2 if $$d_G(u, v) = 1$$dG(u,v)=1 and $$|f(u)-f(v)| \ge 1$$|f(u)-f(v)|≥1 if $$d_G(u, v) = 2$$dG(u,v)=2. The span of f is the difference between the largest and the smallest number in f(V). The $$\lambda $$λ-number $$\lambda (G)$$λ(G) of G is the minimum span over all L(2, 1)-labelings of G. In this paper, we conclude that the $$\lambda $$λ-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).