On the L(2, 1)-labeling conjecture for brick product graphs

Abstract

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).