$L(2,1)$-labeling 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 $f: V \rightarrow \{0,1,\cdots \}$ 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, 2013) in the case when (i) $\ell$ is even, or (ii) $\ell \ge 5$ is odd and $0 \le r \le 8$.