Unavoidable Patterns in Complete Simple Topological Graphs

IF 0.6 3区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Discrete & Computational Geometry Pub Date : 2024-05-29 DOI:10.1007/s00454-024-00658-6

Andrew Suk, Ji Zeng

引用次数: 0

Abstract

We show that every complete n-vertex simple topological graph contains a topological subgraph on at least \((\log n)^{1/4 - o(1)}\) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound \(\Omega (\log ^{1/8}n)\) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least \((\log n)^{1 -o(1)}\).

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完整简单拓扑图中不可避免的模式

我们证明了每一个完整的 n 个顶点的简单拓扑图都包含一个至少 \((\log n)^{1/4 - o(1)}\) 个顶点的拓扑子图，它与完整的凸几何图或完整的扭曲图具有弱同构性。这是对 Pach、Solymosi 和 Tóth 于 2003 年得到的边界 \(\Omega (\log ^{1/8}n)\) 的首次改进。我们还证明了每一个完整的 n 顶点简单拓扑图都包含一条长度至少为 \((\log n)^{1 -o(1)}\) 的平面路径。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discrete & Computational Geometry 数学-计算机：理论方法

CiteScore

1.80

自引率

12.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.