The Independence Ratio of 4-Cycle-Free Planar Graphs

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2026-04-13 Epub Date: 2026-03-15 DOI:10.1002/jgt.70020

Tom Kelly, Sid Kolichala, Caleb McFarland, Jatong Su

{"title":"The Independence Ratio of 4-Cycle-Free Planar Graphs","authors":"Tom Kelly, Sid Kolichala, Caleb McFarland, Jatong Su","doi":"10.1002/jgt.70020","DOIUrl":null,"url":null,"abstract":"We prove that every <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n </semantics></math>-vertex planar graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n </semantics></math> with no triangle sharing an edge with a 4-cycle has independence ratio <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n \n <mo>∕</mo>\n \n <mi>α</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mn>4</mn>\n \n <mo>−</mo>\n \n <mi>ε</mi>\n </mrow>\n </mrow>\n </semantics></math> for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ε</mi>\n \n <mo>=</mo>\n \n <mn>1</mn>\n \n <mo>∕</mo>\n \n <mn>30</mn>\n </mrow>\n </mrow>\n </semantics></math>. This result implies that the same bound holds for 4-cycle-free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5-cycle. For the latter case we strengthen the bound to <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ε</mi>\n \n <mo>=</mo>\n \n <mn>2</mn>\n \n <mo>∕</mo>\n \n <mn>9</mn>\n </mrow>\n </mrow>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"112 2","pages":"116-127"},"PeriodicalIF":1.0000,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70020","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.70020","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/3/15 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We prove that every $n$ -vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n ∕ α (G) \leq 4 - ε$ for $ε = 1 ∕ 30$ . This result implies that the same bound holds for 4-cycle-free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5-cycle. For the latter case we strengthen the bound to $ε = 2 ∕ 9$ .

Abstract Image

查看原文本刊更多论文

无4环平面图的独立比

证明了每一个n顶点的平面图G都具有独立比，且不存在三角形共用一条4环的边n∕α (G)≤4−对于ε = 1∕30。这一结果表明，对于无4环的平面图和无相邻三角形的平面图以及没有三角形与5环共用一条边的平面图，都有相同的界。对于后一种情况，我们将约束强化为ε = 2∕9。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .