{"title":"Tight bound on the minimum degree to guarantee graphs forbidding some odd cycles to be bipartite","authors":"Xiaoli Yuan, Yuejian Peng","doi":"10.1016/j.ejc.2025.104143","DOIUrl":null,"url":null,"abstract":"<div><div>Erdős and Simonovits asked the following question: For an integer <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and a family of non-bipartite graphs <span><math><mi>H</mi></math></span>, determine the infimum of <span><math><mi>α</mi></math></span> such that any <span><math><mi>H</mi></math></span>-free <span><math><mi>n</mi></math></span>-vertex graph with minimum degree at least <span><math><mrow><mi>α</mi><mi>n</mi></mrow></math></span> has chromatic number at most <span><math><mi>r</mi></math></span>. We answer this question for <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span> and any family consisting of odd cycles. Let <span><math><mi>C</mi></math></span> be a family of odd cycles in which <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the shortest odd cycle not in <span><math><mi>C</mi></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the longest odd cycle in <span><math><mi>C</mi></math></span>, we show that if <span><math><mi>G</mi></math></span> is an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>C</mi></math></span>-free graph with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1000</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>8</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>></mo><mo>max</mo><mrow><mo>{</mo><mi>n</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mn>2</mn><mi>n</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>, then <span><math><mi>G</mi></math></span> is bipartite. Moreover, this bound on the minimum degree is tight.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104143"},"PeriodicalIF":1.0000,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669825000253","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Erdős and Simonovits asked the following question: For an integer and a family of non-bipartite graphs , determine the infimum of such that any -free -vertex graph with minimum degree at least has chromatic number at most . We answer this question for and any family consisting of odd cycles. Let be a family of odd cycles in which is the shortest odd cycle not in and is the longest odd cycle in , we show that if is an -vertex -free graph with and , then is bipartite. Moreover, this bound on the minimum degree is tight.
期刊介绍:
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.