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

TL;DR

This paper determines the optimal $L(2,1)$-labeling span for certain strong graph bundles over cycles, specifically when the fiber size is a multiple of 11 and the bundle parameter follows specific modular conditions.

Contribution

It provides exact $ ext{lambda}$-numbers for a class of strong graph bundles over cycles under specific modular conditions, extending previous labeling results.

Findings

$ ext{lambda}$-number equals 10 for specified graph bundles

Conditions involve fiber size multiple of 11 and modular constraints on $\\ell$

Results contribute to graph labeling theory for complex graph structures

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 $\lambda$-number of $G$, denoted by $\lambda (G)$, is the minimum span over all $L(2,1)$-labelings of $G$. We prove that if $X= C_m\boxtimes^{\sigma_\ell} C_{n}$ 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 $\lambda (X)=10$.