Embedding clique subdivisions via crux

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-02-10 DOI:10.1112/jlms.70073

Donglei Yang, Fan Yang

{"title":"Embedding clique subdivisions via crux","authors":"Donglei Yang, Fan Yang","doi":"10.1112/jlms.70073","DOIUrl":null,"url":null,"abstract":"For a graph <math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> with average degree <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$d(G)$</annotation>\n </semantics></math> and a constant <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\alpha >0$</annotation>\n </semantics></math>, we denote by <math>\n <semantics>\n <mrow>\n <msub>\n <mi>C</mi>\n <mi>α</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$C_{\\alpha }(G)$</annotation>\n </semantics></math> the minimum order of a subgraph <math>\n <semantics>\n <mrow>\n <mi>H</mi>\n <mo>⊆</mo>\n <mi>G</mi>\n </mrow>\n <annotation>$H\\subseteq G$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>(</mo>\n <mi>H</mi>\n <mo>)</mo>\n <mo>⩾</mo>\n <mi>α</mi>\n <mi>d</mi>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$d(H)\\geqslant \\alpha d(G)$</annotation>\n </semantics></math>. Liu and Montgomery conjectured that every graph <math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> contains <math>\n <semantics>\n <msub>\n <mi>K</mi>\n <mrow>\n <mi>Ω</mi>\n <mo>(</mo>\n <mi>t</mi>\n <mo>)</mo>\n </mrow>\n </msub>\n <annotation>$K_{\\Omega (t)}$</annotation>\n </semantics></math> as a subdivision for <math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>=</mo>\n <mi>min</mi>\n <mo>{</mo>\n <mi>d</mi>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n <mo>,</mo>\n <msqrt>\n <mstyle>\n <mfrac>\n <mrow>\n <msub>\n <mi>C</mi>\n <mi>α</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <mrow>\n <mi>log</mi>\n <msub>\n <mi>C</mi>\n <mi>α</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n </mfrac>\n </mstyle>\n </msqrt>\n <mo>}</mo>\n </mrow>\n <annotation>$t=\\min \\lbrace d(G), \\sqrt {\\tfrac{C_{\\alpha }(G)}{\\log C_{\\alpha }(G)}}\\rbrace$</annotation>\n </semantics></math>. In the paper, we prove this conjecture.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 2","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70073","RegionNum":2,"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

For a graph $G$ with average degree $d (G)$ and a constant $α > 0$ , we denote by $C_{α} (G)$ the minimum order of a subgraph $H \subseteq G$ with $d (H) ⩾ α d (G)$ . Liu and Montgomery conjectured that every graph $G$ contains $K_{Ω (t)}$ as a subdivision for $t = \min {d (G), \sqrt{\frac{C_{α} (G)}{\log C_{α} (G)}}}$ . In the paper, we prove this conjecture.

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.