Optimal $L(2,1)$-labeling of certain strong graph bundles cycles over cycles

Abstract

An $L(2,1)$-labeling of a graph $G=(V,E)$ is a function $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two, and the labels on vertices at distance two differ by at least one. The span of $f$ is the difference between the largest and the smallest numbers of $f(V)$. The $λ$-number of $G$, denoted by $λ(G)$, is the minimum span over all $L(2,1)$-labelings of $G$. We prove that if $X= C_m\boxtimes^{σ_\ell} C_{n}$ is a direct graph bundle with fiber $C_{n}$ and base $C_m$, $n$ is a multiple of 11 and $\ell$ has a form of $\ell =[11k+(-1)^a 4m]\mod n$ or of $\ell =[11k+(-1)^a 3m]\mod n$, where $a\in \{1,2\}$ and $k\in \ZZ$, then $λ(X)=10$.