{"title":"Polynomial bounds for chromatic number VIII. Excluding a path and a complete multipartite graph","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"10.1002/jgt.23129","DOIUrl":null,"url":null,"abstract":"<p>We prove that for every path <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math>, and every integer <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n </mrow>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math>, there is a polynomial <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>f</mi>\n </mrow>\n </mrow>\n <annotation> $f$</annotation>\n </semantics></math> such that every graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with chromatic number greater than <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>f</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>t</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $f(t)$</annotation>\n </semantics></math> either contains <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> as an induced subgraph, or contains as a subgraph the complete <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n </mrow>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math>-partite graph with parts of cardinality <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>t</mi>\n </mrow>\n </mrow>\n <annotation> $t$</annotation>\n </semantics></math>. For <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>t</mi>\n \n <mo>=</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> $t=1$</annotation>\n </semantics></math> and general <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n </mrow>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math> this is a classical theorem of Gyárfás, and for <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n \n <mo>=</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> $d=2$</annotation>\n </semantics></math> and general <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>t</mi>\n </mrow>\n </mrow>\n <annotation> $t$</annotation>\n </semantics></math> this is a theorem of Bonamy et al.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"509-521"},"PeriodicalIF":0.9000,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23129","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23129","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that for every path , and every integer , there is a polynomial such that every graph with chromatic number greater than either contains as an induced subgraph, or contains as a subgraph the complete -partite graph with parts of cardinality . For and general this is a classical theorem of Gyárfás, and for and general this is a theorem of Bonamy et al.
期刊介绍:
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .