关于图表，拉姆齐擅长写大部头的书

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":"关于图表，拉姆齐擅长写大部头的书","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":"{\"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}","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

摘要

Let G< 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></mrow></ mrow></mrow></ mrow></mrow></math&gt；是一个图形和H&lt； 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></mrow></ mrow></mrow></ mrow></mrow></math&gt；连通图。然后H&lt； 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></mrow></ mrow></mrow></ mrow></math&gt；据称是G&lt； 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></mrow></ mrow></mrow></ mrow></math&gt；-如果拉姆齐数r (G)H) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0005”威利:位置= "方程/ jgt23193 -数学- 0005. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> r< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> G< / mi> & lt; mo>, & lt; / mo> & lt; mi> H< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>等于一般下界（χ (G)）−1)(∣h∣−1) + s (G) <math xmlns=“http://www.w3.org/1998/Math/MathML”altimg = " urn: x-wiley: 03649024:媒体:jgt23193: jgt23193 -数学- 0006“威利:位置= "方程/ jgt23193 -数学- 0006。 png”> < mrow&gt < mrow> &lt mi> s&lt / mi> < mrow&gt < mo&gt (< / mo&gt < mi> G&lt / mi> < mo&gt) < / mo> &lt / mrow> &lt mo> = < / mo&gt < mn> 1&lt / mn> < / mrow&gt < / mrow> &lt / math>,它显示，如果p≥2 &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0013”魏:地方= "方程/ jgt23193-math-0013.png > < mrow&gt < mrow> &lt mi> p&lt / mi> < mo&gt \ unicode {x02265 < / mo> &lt mn> 2< / mn&gt < / mrow> &lt / mrow> &lt / math>,math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0014”魏:地方= "方程/ jgt23193-math-0014.png > < mrow&gt < mrow> &lt msub> < mi&gt K< / mi> &lt mi> p&lt / mi> &lt / msub> < mo&gt \ unicode {x0002B < / mo> &lt mi> n< / mi&gt < mi> H&lt / mi> < / mrow&gt < / mrow> &lt / math>are G< 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>&lt；-good if and only if G< 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>&lt；是M M + K K−1的子图(m) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23193:jgt23193- math0017 ”魏:地方= "方程/ jgt23193-math-0017.png > < mrow&gt < mrow> &lt msub> < mi&gt M< / mi> &lt mi> M&lt / mi> &lt / msub> < mo&gt \ unicode {x0002B < / mo> < msub&gt < mi> K&lt / mi> < mrow&gt < mi> K&lt / mi> < mo&gt \ unicode {x02212 < / mo> &lt mn> 1< / mn&gt < / mrow> &lt / msub> &lt mrow> < mo&gt (< / mo&gt < mi> M&lt / mi> < mo&gt) < / mo> &lt / mrow> < / mrow&gt < / mrow> &lt / math>对于一些m&lt； 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>&lt；,其中M &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23193:jgt23193-math-0019”魏:地方= "方程/ jgt23193-math-0019.png > < mrow&gt < mrow> &lt msub> < mi&gt M< / mi> &lt mi> M< / mi&gt < / msub> &lt / mrow> < / mrow&gt &lt / math>是m &lt；math xmlns="http://www.w3的匹配。

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On graphs for which large books are Ramsey good

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 .