{"title":"On degree powers and counting stars in F-free graphs","authors":"Dániel Gerbner","doi":"10.1016/j.ejc.2025.104135","DOIUrl":null,"url":null,"abstract":"<div><div>Given a positive integer <span><math><mi>r</mi></math></span> and a graph <span><math><mi>G</mi></math></span> with degree sequence <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, we define <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></math></span>. We let <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> be the largest value of <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> if <span><math><mi>G</mi></math></span> is an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graph. We show that if <span><math><mi>F</mi></math></span> has a color-critical edge, then <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for a complete <span><math><mrow><mo>(</mo><mi>χ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-partite graph <span><math><mi>G</mi></math></span> (this was known for cliques and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>). We obtain exact results for several other non-bipartite graphs and also determine <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>r</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. We also give simple proofs of multiple known results.</div><div>Our key observation is the connection to <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, which is the largest number of copies of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graphs, where <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> is the star with <span><math><mi>r</mi></math></span> leaves. We explore this connection and apply methods from the study of <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> to prove our results. We also obtain several new results on <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"126 ","pages":"Article 104135"},"PeriodicalIF":1.0000,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669825000174","RegionNum":3,"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
Given a positive integer and a graph with degree sequence , we define . We let be the largest value of if is an -vertex -free graph. We show that if has a color-critical edge, then for a complete -partite graph (this was known for cliques and ). We obtain exact results for several other non-bipartite graphs and also determine for . We also give simple proofs of multiple known results.
Our key observation is the connection to , which is the largest number of copies of in -vertex -free graphs, where is the star with leaves. We explore this connection and apply methods from the study of to prove our results. We also obtain several new results on .
期刊介绍:
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.