{"title":"Common graphs with arbitrary connectivity and chromatic number","authors":"Sejin Ko , Joonkyung Lee","doi":"10.1016/j.jctb.2023.06.001","DOIUrl":null,"url":null,"abstract":"<div><p>A graph <em>H</em> is <em>common</em> if the number of monochromatic copies of <em>H</em> in a 2-edge-colouring of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is asymptotically minimised by the random colouring. We prove that, given <span><math><mi>k</mi><mo>,</mo><mi>r</mi><mo>></mo><mn>0</mn></math></span>, there exists a <em>k</em><span>-connected common graph with chromatic number at least </span><em>r</em>. The result is built upon the recent breakthrough of Kráľ, Volec, and Wei who obtained common graphs with arbitrarily large chromatic number and answers a question of theirs.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S009589562300045X","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 6
Abstract
A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph is asymptotically minimised by the random colouring. We prove that, given , there exists a k-connected common graph with chromatic number at least r. The result is built upon the recent breakthrough of Kráľ, Volec, and Wei who obtained common graphs with arbitrarily large chromatic number and answers a question of theirs.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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