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{"title":"具有大弧强连通性的超欧拉图","authors":"Jia Wei, Hong-Jian Lai","doi":"10.1002/jgt.23254","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>An oriented graph is a digraph whose underlying graph is simple. Bang-Jensen and Thomassé conjectured that every digraph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n </mrow>\n </mrow>\n </semantics></math> with arc-strong connectivity <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>λ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>D</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math> at least as large as its independence number <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>α</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>D</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math> must be supereulerian. We introduce max–min ditrails in a digraph and investigate the relationship between the arc-strong connectivity <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>λ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>D</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math> and matching number <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>α</mi>\n \n <mo>′</mo>\n </msup>\n \n <mrow>\n <mo>(</mo>\n \n <mi>D</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n </semantics></math> to assure the supereulericity of an oriented graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n </mrow>\n </mrow>\n </semantics></math>. Utilizing the max–min ditrails with the related counting arguments, it is proved that every oriented graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n </mrow>\n </mrow>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>λ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>D</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mrow>\n <mo>⌊</mo>\n \n <mfrac>\n <mrow>\n <msup>\n <mi>α</mi>\n \n <mo>′</mo>\n </msup>\n \n <mrow>\n <mo>(</mo>\n \n <mi>D</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mn>2</mn>\n </mfrac>\n \n <mo>⌋</mo>\n </mrow>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n </semantics></math> is supereulerian. This bound is the best possible.</p>\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 3","pages":"298-312"},"PeriodicalIF":1.0000,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Supereulerian Oriented Graphs With Large Arc-Strong Connectivity\",\"authors\":\"Jia Wei, Hong-Jian Lai\",\"doi\":\"10.1002/jgt.23254\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>\\n \\n <p>An oriented graph is a digraph whose underlying graph is simple. Bang-Jensen and Thomassé conjectured that every digraph <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>D</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> with arc-strong connectivity <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>λ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>D</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math> at least as large as its independence number <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>α</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>D</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math> must be supereulerian. We introduce max–min ditrails in a digraph and investigate the relationship between the arc-strong connectivity <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>λ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>D</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math> and matching number <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msup>\\n <mi>α</mi>\\n \\n <mo>′</mo>\\n </msup>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>D</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mrow>\\n </semantics></math> to assure the supereulericity of an oriented graph <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>D</mi>\\n </mrow>\\n </mrow>\\n </semantics></math>. Utilizing the max–min ditrails with the related counting arguments, it is proved that every oriented graph <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>D</mi>\\n </mrow>\\n </mrow>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>λ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>D</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>≥</mo>\\n \\n <mrow>\\n <mo>⌊</mo>\\n \\n <mfrac>\\n <mrow>\\n <msup>\\n <mi>α</mi>\\n \\n <mo>′</mo>\\n </msup>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>D</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n \\n <mn>2</mn>\\n </mfrac>\\n \\n <mo>⌋</mo>\\n </mrow>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </mrow>\\n </semantics></math> is supereulerian. This bound is the best possible.</p>\\n </div>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"110 3\",\"pages\":\"298-312\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23254\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23254","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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