{"title":"Maximal Degrees in Subgraphs of Kneser Graphs","authors":"Peter Frankl, Andrey Kupavskii","doi":"10.1002/jgt.23213","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>In this paper, we study the maximum degree in nonempty-induced subgraphs of the Kneser graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>KG</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23213:jgt23213-math-0001\" wiley:location=\"equation/jgt23213-math-0001.png\"><mrow><mrow><mi>KG</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>. One of the main results asserts that, for <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>></mo>\n \n <msub>\n <mi>k</mi>\n \n <mn>0</mn>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23213:jgt23213-math-0002\" wiley:location=\"equation/jgt23213-math-0002.png\"><mrow><mrow><mi>k</mi><mo>\\unicode{x0003E}</mo><msub><mi>k</mi><mn>0</mn></msub></mrow></mrow></math></annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n \n <mo>></mo>\n \n <mn>64</mn>\n \n <msup>\n <mi>k</mi>\n \n <mn>2</mn>\n </msup>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23213:jgt23213-math-0003\" wiley:location=\"equation/jgt23213-math-0003.png\"><mrow><mrow><mi>n</mi><mo>\\unicode{x0003E}</mo><mn>64</mn><msup><mi>k</mi><mn>2</mn></msup></mrow></mrow></math></annotation>\n </semantics></math>, whenever a nonempty subgraph has <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>m</mi>\n \n <mo>≥</mo>\n \n <mi>k</mi>\n \n <mfenced>\n <mfrac>\n <mrow>\n <mi>n</mi>\n \n <mo>−</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>2</mn>\n </mrow>\n </mfrac>\n </mfenced>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23213:jgt23213-math-0004\" wiley:location=\"equation/jgt23213-math-0004.png\"><mrow><mrow><mi>m</mi><mo>\\unicode{x02265}</mo><mi>k</mi><mfenced close=\")\" open=\"(\"><mfrac linethickness=\"0\"><mrow><mi>n</mi><mo>\\unicode{x02212}</mo><mn>2</mn></mrow><mrow><mi>k</mi><mo>\\unicode{x02212}</mo><mn>2</mn></mrow></mfrac></mfenced></mrow></mrow></math></annotation>\n </semantics></math> vertices, its maximum degree is at least <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mfrac>\n <mn>1</mn>\n \n <mn>2</mn>\n </mfrac>\n \n <mfenced>\n <mrow>\n <mn>1</mn>\n \n <mo>−</mo>\n \n <mfrac>\n <msup>\n <mi>k</mi>\n \n <mn>2</mn>\n </msup>\n \n <mi>n</mi>\n </mfrac>\n </mrow>\n </mfenced>\n \n <mi>m</mi>\n \n <mo>−</mo>\n \n <mfenced>\n <mfrac>\n <mrow>\n <mi>n</mi>\n \n <mo>−</mo>\n \n <mn>2</mn>\n </mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>2</mn>\n </mrow>\n </mfrac>\n </mfenced>\n \n <mo>≥</mo>\n \n <mn>0.49</mn>\n \n <mi>m</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23213:jgt23213-math-0005\" wiley:location=\"equation/jgt23213-math-0005.png\"><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>\\unicode{x02212}</mo><mfrac><msup><mi>k</mi><mn>2</mn></msup><mi>n</mi></mfrac></mrow></mfenced><mi>m</mi><mo>\\unicode{x02212}</mo><mfenced><mfrac linethickness=\"0\"><mrow><mi>n</mi><mo>\\unicode{x02212}</mo><mn>2</mn></mrow><mrow><mi>k</mi><mo>\\unicode{x02212}</mo><mn>2</mn></mrow></mfrac></mfenced><mo>\\unicode{x02265}</mo><mn>0.49</mn><mi>m</mi></mrow></mrow></math></annotation>\n </semantics></math>. This bound is essentially best possible. One of the intermediate steps is to obtain structural results on nonempty subgraphs with small maximum degree.</p>\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"88-96"},"PeriodicalIF":0.9000,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23213","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the maximum degree in nonempty-induced subgraphs of the Kneser graph . One of the main results asserts that, for and , whenever a nonempty subgraph has vertices, its maximum degree is at least . This bound is essentially best possible. One of the intermediate steps is to obtain structural results on nonempty subgraphs with small maximum degree.
期刊介绍:
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
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