极大平面图中的度的跨度——一篇阐述
Spreads of degrees in maximal planar graphs -- an exposition
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- University of Haifa-Oranim（海法奥拉尼姆大学）
- University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学物理学院）
- Faculty of Information Studies, Novo Mesto（新梅斯托信息学院）
- Rudolfovo – Science and Technology Centre Novo Mesto（鲁道尔福沃-新梅斯托科技中心）
- Department of Mathematics, University of Malta（马耳他大学数学系）
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中文总结 AI 辅助
本文阐述图度序列的跨度主题，聚焦极大平面图，结合相关思路渐近确定了所有对应参数下极大平面图的度跨度最小值，涵盖了重复数情形。
中文摘要 AI 辅助
对于图$G$及其顶点子集$B\subseteq V(G)$，$B$的跨度$\u200b\mathrm{sp}(B)\u200b$是$B$中顶点在$G$内的最大度与最小度之差；对于整数$k\geq0$，参数$\u200b\mathrm{sp}(G,k)\u200b$是满足$\u200b\mathrm{sp}(B)\leq k\u200b$的顶点集$B$的最大基数。Caro、Lauri和Zarb提出，需确定阶为$n$的极大平面图上$\u200b\mathrm{sp}(G,k)\u200b$的最小值，记$\u200b\mathrm{MP}(n,\delta,k)\u200b$为阶为$n$、最小度$\u200b\delta\in\{3,4,5\}\u200b$的极大平面图上的该最小值。本文旨在阐述图$G$度序列中的跨度相关主题，重点聚焦极大平面图。我们将配套论文\cite{P1}的通用下界应用于该类图，并结合进一步思路，部分基于Caro-West、Caro-Lauri-Zarb的成果，部分为新内容。特别地，我们对所有满足$\u200b\delta\in\{3,4,5\}\u200b$且$k\geq0$的$(\u200b\delta,k\u200b)$对，渐近确定了$\u200b\mathrm{MP}(n,\delta,k)\u200b$的值，该结果涵盖了$\u200b\mathrm{MP}(n,\delta,0)\u200b$的情形，即Caro和West提出的重复数$\u200b\mathrm{rep}(G)\u200b$。
英文摘要
For a graph $G$ and a set $B\subseteq V(G)$, the spread $\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\geq 0$ the parameter $\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\mathrm{sp}(B)\leq k$. Caro, Lauri and Zarb asked for the minimum of $\mathrm{sp}(G,k)$ over the maximal planar graphs of order $n$; we write $\mathrm{MP}(n,δ,k)$ for this minimum over the maximal planar graphs of order $n$ and minimum degree $δ\in \{3,4,5\}$. This manuscript is intended as an exposition of the subject of spread in the degree sequence of a graph $G$, with emphasis on maximal planar graphs. We apply the general lower bound of the companion paper \cite{P1} to this class, together with further ideas, some based on Caro--West and Caro--Lauri--Zarb and some new. In particular we asymptotically determine the values of $\mathrm{MP}(n,δ,k)$ for every pair $(δ,k)$ with $δ\in \{3,4,5\}$ and $k\geq 0$. This solution recovers the case $\mathrm{MP}(n,δ,0)$, which is the repetition number $\mathrm{rep}(G)$ introduced by Caro and West.
