{"title":"A Fan-type condition for cycles in 1-tough and k-connected (P2 ∪ kP1)-free graphs","authors":"Zhiquan Hu, Jie Wang, Changlong Shen","doi":"10.1016/j.amc.2025.129300","DOIUrl":null,"url":null,"abstract":"<div><div>For a graph <em>G</em>, define <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>min</mi><mo></mo><mspace></mspace><mo>{</mo><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mspace></mspace><mi>S</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span>, where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the set consisting of all independent sets <span><math><mo>{</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></math></span> of <em>G</em> such that some vertex, say <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (<span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span>), is at distance two from every other vertex in it. A graph <em>G</em> is called 1-tough if for each cut set <span><math><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> has no more than <span><math><mo>|</mo><mi>S</mi><mo>|</mo></math></span> components. Recently, Shi and Shan <span><span>[19]</span></span> conjectured that for each integer <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>, being 2<em>k</em>-connected is sufficient for 1-tough <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∪</mo><mi>k</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span>-free graphs to be hamiltonian, which was confirmed by Xu et al. <span><span>[20]</span></span> and Ota and Sanka <span><span>[16]</span></span>, respectively. In this article, we generalize the above results through the following Fan-type theorem: If <em>G</em> is a 1-tough and <em>k</em>-connected <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∪</mo><mi>k</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span>-free graph and satisfies <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>7</mn><mi>k</mi><mo>−</mo><mn>6</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span>, where <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> is an integer, then <em>G</em> is hamiltonian or the Petersen graph.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129300"},"PeriodicalIF":3.5000,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S009630032500027X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
For a graph G, define , where is the set consisting of all independent sets of G such that some vertex, say (), is at distance two from every other vertex in it. A graph G is called 1-tough if for each cut set , has no more than components. Recently, Shi and Shan [19] conjectured that for each integer , being 2k-connected is sufficient for 1-tough -free graphs to be hamiltonian, which was confirmed by Xu et al. [20] and Ota and Sanka [16], respectively. In this article, we generalize the above results through the following Fan-type theorem: If G is a 1-tough and k-connected -free graph and satisfies , where is an integer, then G is hamiltonian or the Petersen graph.
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.