发表机构
Univ. Grenoble Alpes; CNRS; Laboratoire G-SCOP; Institut Fourier（格勒诺布尔阿尔卑斯大学; 法国国家科学研究中心; G-SCOP实验室; 傅里叶研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究正Lin-Lu-Yau曲率图的有限性，推广Lu-Wang结果至禁止子图类，并完整分类5-连通正曲率平面三角剖分，仅五个例子。
AI 中文摘要
在过去几十年中，人们引入了若干图曲率的概念。其中，通过最优输运定义的Ollivier曲率及其Lin-Lu-Yau变体为图提供了曲率概念，这些概念保留了黎曼几何中对应概念的若干特征。该领域的一个经典问题是理解正曲率空间的整体结构。Lu和Wang证明了最小度至少为3的正曲率平面图仅有有限多个。本文受此结果启发，研究两个方向。首先，我们证明对于固定的$2\leq s\leq t$，最小度至少为$s$的正曲率$K_{s,t}$-无子图仅有有限多个。特别地，这推广了Lu和Wang的结果到可嵌入任何固定曲面的图。其次，我们给出正曲率5-连通平面三角剖分的完整分类。特别地，我们证明这样的图仅有五个例子，分别具有12、14、15、16和17个顶点。
英文摘要
Over the past few decades, several notions of graph curvature have been introduced. Among them, Ollivier curvature, defined through optimal transport, and its Lin--Lu--Yau variant provide notions of curvature for graphs which retain several features of their counterpart in Riemannian geometry. A classical problem in this context is to understand the global structure of positively curved spaces. Lu and Wang proved that there are only finitely many positively curved planar graphs of minimum degree at least 3. In this paper, we study two directions motivated by this result. We first show that for fixed $2\leq s\leq t$, there are only finitely many positively curved $K_{s,t}$-minor-free graphs with minimum degree at least $s$. In particular, this generalizes the result of Lu and Wang to graphs embeddable on any fixed surface. We then give a complete classification of positively curved 5-connected plane triangulations. In particular, we show that there are only five examples, on 12, 14, 15, 16, and 17 vertices.
Comments32 pages, 26 figures. v2: minor corrections
