Optimal $L(d,1)$-labeling of certain direct graph bundles cycles over cycles and Cartesian graph bundles cycles over cycles

Abstract

An $L(d,1)$-labeling of a graph $G$ is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least $d$ and those at a distance of two receive labels that differ by at least one, where $d\geq 1$. Let $\lambda^d_1 (G)$ denote the least $\lambda$ such that $G$ admits an $L(d,1)$-labeling using labels from $\{0,1,\ldots , \lambda \}$. We prove that $\lambda^d_1(X)\leq 2d+2$ for certain direct graph bundle $X= C_m\times^{\sigma_\ell} C_n$ and certain Cartesian graph bundle $X= C_m\Box^{\sigma_\ell} C_n$, where $\sigma_\ell$ is a cyclic $\ell$-shift, with equality if $1\leq d\leq 4$.