{"title":"Ramsey Numbers of Multiple Copies of Graphs in a Component","authors":"Caixia Huang, Yuejian Peng, Yiran Zhang","doi":"10.1007/s00373-024-02821-5","DOIUrl":null,"url":null,"abstract":"<p>For a graph <i>G</i>, let <span>\\(R({\\mathcal {C}}(nG))\\)</span> denote the least <i>N</i> such that every 2-colouring of the edges of <span>\\(K_N\\)</span> contains a monochromatic copy of <i>nG</i> in a monochromatic connected subgraph, where <i>nG</i> denotes <i>n</i> vertex disjoint copies of <i>G</i>. Gyárfás and Sárközy (J Graph Theory 83(2):109–119, 2016) showed that <span>\\(R({\\mathcal {C}}(nK_3))=7n-2\\)</span> for <span>\\(n \\ge 2\\)</span>. After that, Roberts (Electron J Comb 24(1):8, 2017)showed that <span>\\(R({\\mathcal {C}}(nK_r))=(r^2-r+1)n-r+1\\)</span> for <span>\\(r \\ge 4\\)</span> and <span>\\(n \\ge R(K_r)\\)</span>, where <span>\\(R(K_r)\\)</span> is the Ramsey number of <span>\\(K_r\\)</span>. In this paper, we determine <span>\\(R({\\mathcal {C}}(nG))\\)</span> for all 4-vertex graphs <i>G</i> without isolated vertices.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"19 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-08-07","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/s00373-024-02821-5","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
For a graph G, let \(R({\mathcal {C}}(nG))\) denote the least N such that every 2-colouring of the edges of \(K_N\) contains a monochromatic copy of nG in a monochromatic connected subgraph, where nG denotes n vertex disjoint copies of G. Gyárfás and Sárközy (J Graph Theory 83(2):109–119, 2016) showed that \(R({\mathcal {C}}(nK_3))=7n-2\) for \(n \ge 2\). After that, Roberts (Electron J Comb 24(1):8, 2017)showed that \(R({\mathcal {C}}(nK_r))=(r^2-r+1)n-r+1\) for \(r \ge 4\) and \(n \ge R(K_r)\), where \(R(K_r)\) is the Ramsey number of \(K_r\). In this paper, we determine \(R({\mathcal {C}}(nG))\) for all 4-vertex graphs G without isolated vertices.
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.