{"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":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-10-04","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\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2023-10-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0095895623000801\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895623000801","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","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:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.