Extremal Problems for a Matching and Any Other Graph

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-12-19 DOI:10.1002/jgt.23210

Xiutao Zhu, Yaojun Chen

{"title":"Extremal Problems for a Matching and Any Other Graph","authors":"Xiutao Zhu, Yaojun Chen","doi":"10.1002/jgt.23210","DOIUrl":null,"url":null,"abstract":"<div>\n \n For a family of graphs <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0001\" wiley:location=\"equation/jgt23210-math-0001.png\"><mrow><mrow><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">\\unicode{x02131}</mi></mrow></mrow></math></annotation>\n </semantics></math>, a graph is called <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0002\" wiley:location=\"equation/jgt23210-math-0002.png\"><mrow><mrow><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">\\unicode{x02131}</mi></mrow></mrow></math></annotation>\n </semantics></math>-free if it does not contain any member of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0003\" wiley:location=\"equation/jgt23210-math-0003.png\"><mrow><mrow><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">\\unicode{x02131}</mi></mrow></mrow></math></annotation>\n </semantics></math> as a subgraph. The generalized Turán number <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>ex</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mi>r</mi>\n </msub>\n \n <mo>,</mo>\n \n <mi>ℱ</mi>\n </mrow>\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:jgt23210:jgt23210-math-0004\" wiley:location=\"equation/jgt23210-math-0004.png\"><mrow><mrow><mtext>ex</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><msub><mi>K</mi><mi>r</mi></msub><mo>,</mo><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">\\unicode{x02131}</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is the maximum number of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mi>r</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0005\" wiley:location=\"equation/jgt23210-math-0005.png\"><mrow><mrow><msub><mi>K</mi><mi>r</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> in an <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0006\" wiley:location=\"equation/jgt23210-math-0006.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math>-vertex <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0007\" wiley:location=\"equation/jgt23210-math-0007.png\"><mrow><mrow><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">\\unicode{x02131}</mi></mrow></mrow></math></annotation>\n </semantics></math>-free graph and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>ex</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>,</mo>\n \n <mi>ℱ</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mtext>ex</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>ℱ</mi>\n </mrow>\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:jgt23210:jgt23210-math-0008\" wiley:location=\"equation/jgt23210-math-0008.png\"><mrow><mrow><mtext>ex</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><msub><mi>K</mi><mn>2</mn></msub><mo>,</mo><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">\\unicode{x02131}</mi></mrow><mo>)</mo></mrow><mo>=</mo><mtext>ex</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">\\unicode{x02131}</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>, that is, the classical Turán number. Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mrow>\n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0009\" wiley:location=\"equation/jgt23210-math-0009.png\"><mrow><mrow><msub><mi>M</mi><mrow><mi>s</mi><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></msub></mrow></mrow></math></annotation>\n </semantics></math> be a matching on <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0010\" wiley:location=\"equation/jgt23210-math-0010.png\"><mrow><mrow><mi>s</mi><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math> edges and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0011\" wiley:location=\"equation/jgt23210-math-0011.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math> be any graph. In this paper, we determine <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>ex</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mi>r</mi>\n </msub>\n \n <mo>,</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mrow>\n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <mi>F</mi>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\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:jgt23210:jgt23210-math-0012\" wiley:location=\"equation/jgt23210-math-0012.png\"><mrow><mrow><mtext>ex</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><msub><mi>K</mi><mi>r</mi></msub><mo>,</mo><mrow><mo class=\"MathClass-open\">{</mo><mrow><msub><mi>M</mi><mrow><mi>s</mi><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>F</mi></mrow><mo class=\"MathClass-close\">}</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> apart from a constant additive term and also give a condition when the error constant term can be determined. In particular, we give the exact value of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>ex</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mrow>\n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <mi>F</mi>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\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:jgt23210:jgt23210-math-0013\" wiley:location=\"equation/jgt23210-math-0013.png\"><mrow><mrow><mtext>ex</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mrow><mo class=\"MathClass-open\">{</mo><mrow><msub><mi>M</mi><mrow><mi>s</mi><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>F</mi></mrow><mo class=\"MathClass-close\">}</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0014\" wiley:location=\"equation/jgt23210-math-0014.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math> being any non-bipartite graph or some bipartite graphs. Furthermore, we determine <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>ex</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mi>r</mi>\n </msub>\n \n <mo>,</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mrow>\n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <mi>F</mi>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\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:jgt23210:jgt23210-math-0015\" wiley:location=\"equation/jgt23210-math-0015.png\"><mrow><mrow><mtext>ex</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><msub><mi>K</mi><mi>r</mi></msub><mo>,</mo><mrow><mo class=\"MathClass-open\">{</mo><mrow><msub><mi>M</mi><mrow><mi>s</mi><mo>\\unicode{x0002B}</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>F</mi></mrow><mo class=\"MathClass-close\">}</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> when <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0016\" wiley:location=\"equation/jgt23210-math-0016.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math> is color critical with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mi>max</mi>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mi>r</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\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:jgt23210:jgt23210-math-0017\" wiley:location=\"equation/jgt23210-math-0017.png\"><mrow><mrow><mi>\\unicode{x003C7}</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>\\unicode{x02265}</mo><mi>max</mi><mrow><mo class=\"MathClass-open\">{</mo><mrow><mi>r</mi><mo>\\unicode{x0002B}</mo><mn>1</mn><mo>,</mo><mn>4</mn></mrow><mo class=\"MathClass-close\">}</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"19-24"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-19","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.23210","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

For a family of graphs $ℱ$ , a graph is called $ℱ$ -free if it does not contain any member of $ℱ$ as a subgraph. The generalized Turán number $ex (n, K_{r}, ℱ)$ is the maximum number of $K_{r}$ in an $n$ -vertex $ℱ$ -free graph and $ex (n, K_{2}, ℱ) = ex (n, ℱ)$ , that is, the classical Turán number. Let $M_{s + 1}$ be a matching on $s + 1$ edges and $F$ be any graph. In this paper, we determine $ex (n, K_{r}, {M_{s + 1}, F})$ apart from a constant additive term and also give a condition when the error constant term can be determined. In particular, we give the exact value of $ex (n, {M_{s + 1}, F})$ for $F$ being any non-bipartite graph or some bipartite graphs. Furthermore, we determine $ex (n, K_{r}, {M_{s + 1}, F})$ when $F$ is color critical with $χ (F) \geq \max {r + 1, 4}$ .

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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 .