关于圈的反极标号的一个猜想

中文摘要 AI 辅助

设 $G$ 表示一个有限、连通、简单的图，$D$ 表示其直径，$d_{G}(u, v)$ 表示 $V(G)$ 中顶点 $u$ 和 $v$ 之间的距离。$G$ 的一个反极标号是映射 $f\colon V(G) \to \mathbb{N}_{0}$，使得对于 $V(G)$ 中每一对不同的顶点 $(u, v)$，关系 $ |f(u) - f(v)| \geq D - d_{G}(u, v) $ 成立。$f$ 的跨度定义为 $\operatorname{sp}(f) = \max\{ f(u) - f(v): u, v \in V(G) \}$。$G$ 的反极数，记作 $\operatorname{an}(G)$，定义为 $G$ 的所有反极标号中可能的最小跨度。Juan 和 Liu [Ars Combin., 2012] 证明了上界 $\operatorname{an}(C_{4k}) \leq 2k^2 - 1$，并猜想对于每个正整数 $k$，有 $\operatorname{an}(C_{4k}) = 2k^2 - 1$，且 Juan 和 Liu 验证了该猜想对 $k \leq 5$ 成立。我们成功证明了 Juan 和 Liu 的猜想，该猜想似乎一直未被解决。多年来，许多作者注意到了 Juan 和 Liu 猜想的开放性，包括 Rao 等人 [Contrib. Discrete Math., 2015] 和 Saha 等人 [Theory Comput. Syst., 2022]。

英文摘要

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 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 span of $f$ is then defined so that $\operatorname{sp}(f) = \max\{ f(u) - f(v) : u, v \in V(G) \}$. The 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 [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. [Contrib. Discrete Math., 2015] and Saha et al. [Theory Comput. Syst., 2022].

发表机构

• Dalhousie University（达尔豪斯大学）

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