On graphs for which large books are Ramsey good

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-10-07 DOI:10.1002/jgt.23193

Meng Liu, Yusheng Li

{"title":"On graphs for which large books are Ramsey good","authors":"Meng Liu, Yusheng Li","doi":"10.1002/jgt.23193","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0001\" wiley:location=\"equation/jgt23193-math-0001.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> be a graph and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0002\" wiley:location=\"equation/jgt23193-math-0002.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> a connected graph. Then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0003\" wiley:location=\"equation/jgt23193-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> is said to be <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0004\" wiley:location=\"equation/jgt23193-math-0004.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math>-good if the Ramsey number <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>G</mi>\n \n <mo>,</mo>\n \n <mi>H</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:jgt23193:jgt23193-math-0005\" wiley:location=\"equation/jgt23193-math-0005.png\"><mrow><mrow><mi>r</mi><mrow><mo>(</mo><mrow><mi>G</mi><mo>,</mo><mi>H</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is equal to the general lower bound <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>H</mi>\n \n <mo>∣</mo>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>+</mo>\n \n <mi>s</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</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:jgt23193:jgt23193-math-0006\" wiley:location=\"equation/jgt23193-math-0006.png\"><mrow><mrow><mrow><mo>(</mo><mrow><mi>\\unicode{x003C7}</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mo>\\unicode{x02223}</mo><mi>H</mi><mo>\\unicode{x02223}</mo><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>\\unicode{x0002B}</mo><mi>s</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>, where <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</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:jgt23193:jgt23193-math-0007\" wiley:location=\"equation/jgt23193-math-0007.png\"><mrow><mrow><mi>\\unicode{x003C7}</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</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:jgt23193:jgt23193-math-0008\" wiley:location=\"equation/jgt23193-math-0008.png\"><mrow><mrow><mi>s</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> are the chromatic number and the chromatic surplus of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0009\" wiley:location=\"equation/jgt23193-math-0009.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math>, respectively. For a fixed graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0010\" wiley:location=\"equation/jgt23193-math-0010.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0011\" wiley:location=\"equation/jgt23193-math-0011.png\"><mrow><mrow><mi>\\unicode{x003C7}</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi><mo>\\unicode{x0002B}</mo><mn>1</mn><mo>\\unicode{x02265}</mo><mn>2</mn></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\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:jgt23193:jgt23193-math-0012\" wiley:location=\"equation/jgt23193-math-0012.png\"><mrow><mrow><mi>s</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math>, it is shown that if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>p</mi>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0013\" wiley:location=\"equation/jgt23193-math-0013.png\"><mrow><mrow><mi>p</mi><mo>\\unicode{x02265}</mo><mn>2</mn></mrow></mrow></math></annotation>\n </semantics></math>, then large <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mi>p</mi>\n </msub>\n \n <mo>+</mo>\n \n <mi>n</mi>\n \n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0014\" wiley:location=\"equation/jgt23193-math-0014.png\"><mrow><mrow><msub><mi>K</mi><mi>p</mi></msub><mo>\\unicode{x0002B}</mo><mi>n</mi><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> are <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0015\" wiley:location=\"equation/jgt23193-math-0015.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math>-good if and only if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0016\" wiley:location=\"equation/jgt23193-math-0016.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> is a subgraph of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mi>m</mi>\n </msub>\n \n <mo>+</mo>\n \n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>k</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>m</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:jgt23193:jgt23193-math-0017\" wiley:location=\"equation/jgt23193-math-0017.png\"><mrow><mrow><msub><mi>M</mi><mi>m</mi></msub><mo>\\unicode{x0002B}</mo><msub><mi>K</mi><mrow><mi>k</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> for some <math>\n <semantics>\n <mrow>\n \n <mrow>\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:jgt23193:jgt23193-math-0018\" wiley:location=\"equation/jgt23193-math-0018.png\"><mrow><mrow><mi>m</mi></mrow></mrow></math></annotation>\n </semantics></math>, where <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>M</mi>\n \n <mi>m</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0019\" wiley:location=\"equation/jgt23193-math-0019.png\"><mrow><mrow><msub><mi>M</mi><mi>m</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> is a matching of <math>\n <semantics>\n <mrow>\n \n <mrow>\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:jgt23193:jgt23193-math-0020\" wiley:location=\"equation/jgt23193-math-0020.png\"><mrow><mrow><mi>m</mi></mrow></mrow></math></annotation>\n </semantics></math> edges. We also give conditions for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0021\" wiley:location=\"equation/jgt23193-math-0021.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> with respect to <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0022\" wiley:location=\"equation/jgt23193-math-0022.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math>-goodness of large <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>+</mo>\n \n <mi>n</mi>\n \n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0023\" wiley:location=\"equation/jgt23193-math-0023.png\"><mrow><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>\\unicode{x0002B}</mo><mi>n</mi><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"543-559"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-07","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.23193","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

Let $G$ be a graph and $H$ a connected graph. Then $H$ is said to be $G$ -good if the Ramsey number $r (G, H)$ is equal to the general lower bound $(χ (G) - 1) (∣ H ∣ - 1) + s (G)$ , where $χ (G)$ and $s (G)$ are the chromatic number and the chromatic surplus of $G$ , respectively. For a fixed graph $G$ with $χ (G) = k + 1 \geq 2$ and $s (G) = 1$ , it is shown that if $p \geq 2$ , then large $K_{p} + n H$ are $G$ -good if and only if $G$ is a subgraph of $M_{m} + K_{k - 1} (m)$ for some $m$ , where $M_{m}$ is a matching of $m$ edges. We also give conditions for $G$ with respect to $G$ -goodness of large $K_{1} + n H$ .

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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 .