{"title":"Clustered coloring of (path + 2K1)-free graphs on surfaces","authors":"Zdeněk Dvořák","doi":"10.1016/j.jctb.2025.02.001","DOIUrl":null,"url":null,"abstract":"<div><div>Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>″</mo></mrow></msubsup></math></span> be the complete join of two isolated vertices with a path on <em>t</em> vertices. For any surface Σ, a subgraph-closed class of graphs drawn on Σ is 3-choosable with bounded clustering if and only if there exists <em>t</em> such that <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>″</mo></mrow></msubsup></math></span> does not belong to the class.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 45-67"},"PeriodicalIF":1.2000,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series B","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895625000085","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let be the complete join of two isolated vertices with a path on t vertices. For any surface Σ, a subgraph-closed class of graphs drawn on Σ is 3-choosable with bounded clustering if and only if there exists t such that does not belong to the class.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.