Year: 2026
Type: preprint
Abstract: Let $G$ denote a finite, connected, simple graph, and let $D$ denote its diameter, and let $d_{G}(u, v)$ denote the distance between vertices $u$ and $v$ in $V(G)$. An \emph{antipodal labeling} of $G$ is a mapping $f\colon V(G) \to \mathbb{N}_{0}$ such that, for each pair $(u, v)$ consisting of distinct vertices in $V(G)$, the relation $ |f(u) - f(v)| \geq D - d_{G}(u, v) $ holds. The \emph{span} of $f$ is then defined so that $\operatorname{sp}(f) = \max\{ f(u) - f(v) : u, v \in V(G) \}$. The \emph{antipodal number} of $G$, denoted with $\operatorname{an}(G)$, may then be defined as the minimum possible span among all antipodal labelings of $G$. Juan and Liu [\emph{Ars Combin.}, 2012] proved the upper bound $\operatorname{an}(C_{4k}) \leq 2k^2 - 1$, and conjectured that $\operatorname{an}(C_{4k}) = 2k^2 - 1$ for each positive integer $k$, and Juan and Liu verified this conjecture for $k \leq 5$. We succeed in proving Juan and Liu's conjecture, which seems to have remained open. The openness of Juan and Liu's conjecture has been noted, over the years, by many authors, including Rao et al.\ [\emph{Contrib. Discrete Math.}, 2015] and Saha et al.\ [\emph{Theory Comput. Syst.}, 2022].
Source: arXiv (Cornell University)
Author: Campbell, John M.
Institution: Dalhousie University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204006435