{"title":"A Note on the Gyárfás–Sumner Conjecture","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"10.1007/s00373-024-02754-z","DOIUrl":null,"url":null,"abstract":"<p>The Gyárfás–Sumner conjecture says that for every tree <i>T</i> and every integer <span>\\(t\\ge 1\\)</span>, if <i>G</i> is a graph with no clique of size <i>t</i> and with sufficiently large chromatic number, then <i>G</i> contains an induced subgraph isomorphic to <i>T</i>. This remains open, but we prove that under the same hypotheses, <i>G</i> contains a subgraph <i>H</i> isomorphic to <i>T</i> that is “path-induced”; that is, for some distinguished vertex <i>r</i>, every path of <i>H</i> with one end <i>r</i> is an induced path of <i>G</i>.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"37 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Graphs and Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02754-z","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The Gyárfás–Sumner conjecture says that for every tree T and every integer \(t\ge 1\), if G is a graph with no clique of size t and with sufficiently large chromatic number, then G contains an induced subgraph isomorphic to T. This remains open, but we prove that under the same hypotheses, G contains a subgraph H isomorphic to T that is “path-induced”; that is, for some distinguished vertex r, every path of H with one end r is an induced path of G.
Gyárfás-Sumner 猜想说，对于每棵树 T 和每个整数 \（t\ge 1\），如果 G 是一个没有大小为 t 的簇且色度数足够大的图，那么 G 包含一个与 T 同构的诱导子图。这一点仍未解决，但我们证明，在同样的假设下，G 包含一个与 T 同构的子图 H，它是 "路径诱导 "的；也就是说，对于某个区分顶点 r，H 的每条路径的一个端点 r 都是 G 的一条诱导路径。
期刊介绍：
Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers, the journal also features survey articles from authors invited by the editorial board.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
