Characterization of minimally t-tough, 2K2-free graphs for 1 IF 1 3区 数学 Q3 MATHEMATICS, APPLIED Discrete Applied Mathematics Pub Date : 2025-07-04 DOI:10.1016/j.dam.2025.06.045 Hui Ma, Xiaomin Hu, Weihua Yang 求助PDF {"title":"Characterization of minimally t-tough, 2K2-free graphs for 1<t≤2","authors":"Hui Ma,&nbsp;Xiaomin Hu,&nbsp;Weihua Yang","doi":"10.1016/j.dam.2025.06.045","DOIUrl":null,"url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> is minimally <span><math><mi>t</mi></math></span>-tough if the toughness of <span><math><mi>G</mi></math></span> is exactly <span><math><mi>t</mi></math></span> and the removal of any edge decreases the toughness. Kriesell’s conjecture, stating that every minimally 1-tough graph has a vertex of degree 2, is still open for general graphs. Katona and Varga generalized Kriesell’s conjecture that every minimally <span><math><mi>t</mi></math></span>-tough graph has a vertex of degree <span><math><mrow><mo>⌈</mo><mn>2</mn><mi>t</mi><mo>⌉</mo></mrow></math></span> for any positive rational number <span><math><mi>t</mi></math></span>. We have confirmed Kriesell’s conjecture for <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graphs by showing that every minimally 1-tough, <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graph is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>. In this paper, we prove that for <span><math><mrow><mn>1</mn><mo>&lt;</mo><mi>t</mi><mo>≤</mo><mn>2</mn></mrow></math></span>, every minimally <span><math><mi>t</mi></math></span>-tough, <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graph on at least 14 vertices has a vertex of degree <span><math><mrow><mo>⌈</mo><mn>2</mn><mi>t</mi><mo>⌉</mo></mrow></math></span> by characterizing the structure of these graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 43-50"},"PeriodicalIF":1.0000,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X25003567","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0} 引用次数: 0 引用 批量引用 Abstract A graph G is minimally t-tough if the toughness of G is exactly t and the removal of any edge decreases the toughness. Kriesell’s conjecture, stating that every minimally 1-tough graph has a vertex of degree 2, is still open for general graphs. Katona and Varga generalized Kriesell’s conjecture that every minimally t-tough graph has a vertex of degree ⌈2t⌉ for any positive rational number t. We have confirmed Kriesell’s conjecture for 2K2-free graphs by showing that every minimally 1-tough, 2K2-free graph is C4 or C5. In this paper, we prove that for 1<t≤2, every minimally t-tough, 2K2-free graph on at least 14 vertices has a vertex of degree ⌈2t⌉ by characterizing the structure of these graphs. 分享 查看原文 本刊更多论文 1 如果图G的韧性正好为t，并且去掉任何边都会降低韧性，那么图G就是最小t韧性。Kriesell的猜想，即每个最小1-tough图都有一个2度的顶点，对于一般图仍然是开放的。Katona和Varga推广了Kriesell的猜想，即对于任意正有理数t，每个最小t-坚韧图都有一个顶点为≤2t²。我们通过证明每个最小1-坚韧，无2k2的图都是C4或C5，证实了Kriesell关于无2k2图的猜想。本文通过刻画图的结构，证明了对于1<；t≤2，每一个至少有14个顶点的最小t-坚韧，无2k2的图都有一个顶点为≤2t； 本文章由计算机程序翻译，如有差异，请以英文原文为准。 求助全文 约1分钟内获得全文 求助全文 来源期刊 Discrete Applied Mathematics 数学-应用数学 CiteScore 2.30 自引率 9.10% 发文量 422 审稿时长 4.5 months 期刊介绍： The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. 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Abstract

A graph $G$ is minimally $t$-tough if the toughness of $G$ is exactly $t$ and the removal of any edge decreases the toughness. Kriesell’s conjecture, stating that every minimally 1-tough graph has a vertex of degree 2, is still open for general graphs. Katona and Varga generalized Kriesell’s conjecture that every minimally $t$-tough graph has a vertex of degree $\lceil 2t\rceil$ for any positive rational number $t$. We have confirmed Kriesell’s conjecture for $2K_2$-free graphs by showing that every minimally 1-tough, $2K_2$-free graph is $C_4$ or $C_5$. In this paper, we prove that for $1<t\le 2$, every minimally $t$-tough, $2K_2$-free graph on at least 14 vertices has a vertex of degree $\lceil 2t\rceil$ by characterizing the structure of these graphs.