{"title":"Every 2-connected {claw, Z2}-free graph with minimum degree at least 4 contains two CISTs","authors":"Xiaodong Chen , Jiayuan Zhang , Liming Xiong , Guifu Su","doi":"10.1016/j.dam.2024.08.020","DOIUrl":null,"url":null,"abstract":"<div><p>If a graph <span><math><mi>G</mi></math></span> contains two spanning trees <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> such that for each two distinct vertices <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></math></span> of <span><math><mrow><mi>G</mi><mo>,</mo></mrow></math></span> the <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math></span>-path in each <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> has no common edge and no common vertex except for the two ends, then <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> are called two completely independent spanning trees (CISTs) of <span><math><mrow><mi>G</mi><mo>,</mo><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow><mo>.</mo></mrow></math></span> There are several results on the existence of two CISTs. In this paper, we prove that every 2-connected {claw, <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>}-free graph with minimum degree at least 4 contains two CISTs. The bound of the minimum degree in our result is best possible.</p></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 51-64"},"PeriodicalIF":1.0000,"publicationDate":"2024-09-03","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/S0166218X24003780","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
Abstract
If a graph contains two spanning trees such that for each two distinct vertices of the -path in each has no common edge and no common vertex except for the two ends, then are called two completely independent spanning trees (CISTs) of There are several results on the existence of two CISTs. In this paper, we prove that every 2-connected {claw, }-free graph with minimum degree at least 4 contains two CISTs. The bound of the minimum degree in our result is best possible.
如果一个图 G 包含两棵生成树 T1,T2，对于 G 的两个不同顶点 x,y，每个 Ti 中的（x,y）路径除了两端外没有公共边和公共顶点，那么 T1,T2 被称为 G 的两棵完全独立生成树（CIST），i∈{1,2}。关于两棵完全独立生成树的存在性有多种结果。在本文中，我们证明了每一个最小度至少为 4 的 2 连接{claw, Z2} 无图都包含两个 CIST。在我们的结果中，最小度的约束是最好的。
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. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
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