Maximal Degrees in Subgraphs of Kneser Graphs

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Graph Theory Pub Date : 2025-01-08 DOI:10.1002/jgt.23213

Peter Frankl, Andrey Kupavskii

{"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 In this paper, we study the maximum degree in nonempty-induced subgraphs of the Kneser graph <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 <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 <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 <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 <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.\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 $KG (n, k)$ . One of the main results asserts that, for $k > k_{0}$ and $n > 64 k^{2}$ , whenever a nonempty subgraph has $m \geq k (\frac{n - 2}{k - 2})$ vertices, its maximum degree is at least $\frac{1}{2} (1 - \frac{k^{2}}{n}) m - (\frac{n - 2}{k - 2}) \geq 0.49 m$ . This bound is essentially best possible. One of the intermediate steps is to obtain structural results on nonempty subgraphs with small maximum degree.

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

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. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .

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Issue Information Issue Information Issue Information The Complexity of Decomposing a Graph into a Matching and a Bounded Linear Forest Maximal Degrees in Subgraphs of Kneser Graphs