On degree powers and counting stars in F-free graphs

IF 1 3区 数学 Q1 MATHEMATICS
Dániel Gerbner
{"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 r and a graph G with degree sequence d1,,dn, we define er(G)=i=1ndir. We let exr(n,F) be the largest value of er(G) if G is an n-vertex F-free graph. We show that if F has a color-critical edge, then exr(n,F)=er(G) for a complete (χ(F)1)-partite graph G (this was known for cliques and C5). We obtain exact results for several other non-bipartite graphs and also determine exr(n,C4) for r3. We also give simple proofs of multiple known results.
Our key observation is the connection to ex(n,Sr,F), which is the largest number of copies of Sr in n-vertex F-free graphs, where Sr is the star with r leaves. We explore this connection and apply methods from the study of ex(n,Sr,F) to prove our results. We also obtain several new results on ex(n,Sr,F).
论无f图中的次幂和计数星
给定一个正整数r和一个度序列为d1,…,dn的图G,我们定义er(G)=∑i=1ndir。我们设exr(n,F)是er(G)的最大值如果G是一个n顶点的无F图。我们证明了如果F有一条颜色临界边,那么对于完全(χ(F)−1)部图G(这是已知的团和C5),则exr(n,F)=er(G)。我们得到了其他几个非二部图的精确结果,并确定了r≥3时的exr(n,C4)。我们也给出了多个已知结果的简单证明。我们的关键观察是与ex(n,Sr,F)的联系,它是n顶点无F图中Sr的最大拷贝数,其中Sr是具有r个叶子的星形。我们探索了这种联系,并应用了ex(n,Sr,F)的研究方法来证明我们的结果。我们还得到了ex(n,Sr,F)的几个新结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.10
自引率
10.00%
发文量
124
审稿时长
4-8 weeks
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.
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