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运筹学学报 >

2021 , Vol. 25 >Issue 3: 200 - 216

DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2021.03.013

图的平面Turán数和平面anti-Ramsey数

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  • 1. 河北工业大学理学院, 天津 300401;
    2. 南开大学组合数学中心, 天津 300071;
    3. 中佛罗里达大学数学系, 奥兰多 FL 32816, 美国

收稿日期: 2021-03-15

  网络出版日期: 2021-09-26

基金资助

国家自然科学基金（Nos.12001154，11922112，11771221），美国国家科学基金（No.DMS-1854903），天津市自然科学基金（Nos.20JCJQJC00090，20JCZDJC00840），中央高校南开大学基本科研业务费专项资金（No.63213037），天津市共建高校专项资金（No.280000307）

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Planar Turán number and planar anti-Ramsey number of graphs

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  • 1. School of Science, Hebei University of Technology, Tianjin 300401, China;
    2. Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, China;
    3. Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

Received date: 2021-03-15

  Online published: 2021-09-26

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摘要

在所有顶点数为$n$且不包含图$G$作为子图的平面图中，具有最多边数的图的边数称为图$G$的平面Turán数，记为$ex_{_\mathcal{P}}（n，G）$。给定正整数$n$以及平面图$H$，用$\mathcal{T}_n （H）$来表示所有顶点数为$n$且不包含$H$作为子图的平面三角剖分图所组成的图集合。设图集合$\mathcal{T}_n （H）$中的任意平面三角剖分图的任意$k$边染色都不包含彩虹子图$H$，则称满足上述条件的$k$的最大值为图$H$的平面anti-Ramsey数，记作$ar_{_\mathcal{P}}（n，H）$。两类问题的研究均始于2015年左右，至今已经引起了广泛关注。全面地综述两类问题的主要研究成果，以及一些公开问题。

关键词： 平面Turán数; 平面anti-Ramsey数; Theta图

本文引用格式

兰永新, 史永堂, 宋梓霞 . 图的平面Turán数和平面anti-Ramsey数[J]. 运筹学学报, 2021 , 25(3) : 200 -216 . DOI: 10.15960/j.cnki.issn.1007-6093.2021.03.013

Abstract

The planar Turán number of a graph $G$, denoted $ex_{_\mathcal{P}}(n,G)$, is the maximum number of edges in a planar graph on $n$ vertices without containing $G$ as a subgraph. Given a positive integer $n$ and a plane graph $H$, let $\mathcal{T}_n(H)$ be the family of all plane triangulations $T$ on $n$ vertices such that $T$ contains $H$ as a subgraph. The planar anti-Ramsey number of $H$, denoted $ar_{_\mathcal{P}}(n, H)$, is the maximum number $k$ such that no edge-coloring of any plane triangulation in $\mathcal{T}_n(H)$ with $k$ colors contains a rainbow copy of $H$. The study of these two topics was initiated around 2015, and has attracted extensive attention. This paper surveys results about planar Turán number and planar anti-Ramsey number of graphs. The goal is to give a unified and comprehensive presentation of the major results, as well as to highlight some open problems.

Key words： planar Turán number; planar anti-Ramsey number; Theta graph

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