{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02754-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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。
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