{"title":"稠密正则有向图和有向图中的Hamilton环","authors":"Allan Lo , Viresh Patel , Mehmet Akif Yıldız","doi":"10.1016/j.jctb.2023.09.004","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that for every <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> there exists <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>ε</mi><mo>)</mo></math></span> such that every regular oriented graph on <span><math><mi>n</mi><mo>></mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> vertices and degree at least <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>+</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"164 ","pages":"Pages 119-160"},"PeriodicalIF":1.3000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Hamilton cycles in dense regular digraphs and oriented graphs\",\"authors\":\"Allan Lo , Viresh Patel , Mehmet Akif Yıldız\",\"doi\":\"10.1016/j.jctb.2023.09.004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that for every <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> there exists <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>ε</mi><mo>)</mo></math></span> such that every regular oriented graph on <span><math><mi>n</mi><mo>></mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> vertices and degree at least <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>+</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.</p></div>\",\"PeriodicalId\":54865,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series B\",\"volume\":\"164 \",\"pages\":\"Pages 119-160\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series B\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0095895623000801\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2023/10/4 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series B","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895623000801","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/10/4 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Hamilton cycles in dense regular digraphs and oriented graphs
We prove that for every there exists such that every regular oriented graph on vertices and degree at least has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.
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