匹配图和其他图的极值问题

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":"匹配图和其他图的极值问题","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":"{\"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}","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

摘要

对于一组图{&lt；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 - text - igrapigrapic " mathvariant="normal">\unicode{x02131}</ mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></math&gt；，一个图形被称为<s:1> <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23210:jgt23210- jgt23210-math-0002" wiley:location="equation/jgt23210-math-0002.png"><mrow><mrow><mi class=" mjx - text - igrapigrapic " mathvariant="normal">\unicode{x02131}</ mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；如果不包含任何成员则免费：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 - text - igrapigrapic " mathvariant="normal">\unicode{x02131}</ mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；作为子图。广义的Turán数ex (n)，K r，math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0004”威利:位置= "方程/ jgt23210 -数学- 0004. png”祝辞& lt; mrow> & lt; mrow> & lt; mtext> ex< / mtext> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; msub> & lt; mi> K< / mi> & lt; mi> r< / mi> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mi类=“MJX-tex-caligraphic”mathvariant =“正常”祝辞\ unicode {x02131} & lt; / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>是K的最大数量r &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0005”威利:位置= "方程/ jgt23210 -数学- 0005. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> K< / mi> & lt; mi> r< / mi> & lt; / msub> & lt; / mrow> & lt; / mrow> & lt; / math>in an <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>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；-vertex _ (<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. ” mrow><mrow><mi class=" mjx - text - calligraphy " mathvariant="normal">\unicode{x02131}</ mrow></mrow></ mrow></math&gt；自由图和ex (n)k2，(n) = ex (n)；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0008”威利:位置= "方程/ jgt23210 -数学- 0008. png”祝辞& lt; mrow> & lt; mrow> & lt; mtext> ex< / mtext> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; msub> & lt; mi> K< / mi> & lt; mn> 2 & lt; / mn> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mi类=“MJX-tex-caligraphic”mathvariant = "正常"祝辞\ unicode {x02131} & lt; / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; mo> = & lt; / mo> & lt; mtext> ex< / mtext> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; mi类=“MJX-tex-caligraphic”mathvariant =“正常”祝辞\ unicode {x02131} & lt; / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>，即经典的Turán数。设M + 1 &lt；math xmlns=“http://www.w3.org/1998/Math/MathML”altimg = " urn: x-wiley: 03649024:媒体:jgt23210: jgt23210 -数学- 0009“威利:位置=“方程/ jgt23210 -数学- 0009. - png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> M< / mi> & lt; mrow> & lt; mi> s< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; / mrow> & lt; / msub> & lt; / mrow> & lt; / mrow> & lt; / math>在s + 1 &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0010”威利:位置= "方程/ jgt23210 -数学- 0010. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> s< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>edge and F< 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></mrow></ mrow></mrow>&lt；是任意图形。在本文中，我们确定ex (n)，K r，{M s + 1,F}) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0012”威利:位置= "方程/ jgt23210 -数学- 0012. png”祝辞& lt; mrow> & lt; mrow> & lt; mtext> ex< / mtext> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; msub> & lt; mi> K< / mi> & lt; mi> r< / mi> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mrow> & lt;莫类= " MathClass-open "祝辞{& lt; / mo> & lt; mrow> & lt; msub> & lt; mi> M< / mi> & lt; mrow> & lt; mi> s< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; / mrow> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mi> F< / mi> & lt; / mrow> & lt;莫类=“MathClass-close”祝辞}& lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>除了一个常数加性项外，还给出了误差常数项可以确定的条件。特别地，我们给出了ex (n)的确切值，{M s + 1,F}) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0013”威利:位置= "方程/ jgt23210 -数学- 0013. png”祝辞& lt; mrow> & lt; mrow> & lt; mtext> ex< / mtext> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; mrow> & lt;莫类=“MathClass-open祝辞{& lt; / mo> & lt; mrow> & lt; msub> & lt; mi> M< / mi> & lt; mrow> & lt; mi> s< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; / mrow> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mi> F< / mi> & lt; / mrow> & lt;莫类= " MathClass-close "祝辞}& lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>for F< 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>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；是任意非二部图或某些二部图。进一步，我们确定ex (n)，K r，{M s + 1,F}) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0015”威利:位置= "方程/ jgt23210 -数学- 0015. png”祝辞& lt; mrow> & lt; mrow> & lt; mtext> ex< / mtext> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; msub> & lt; mi> K< / mi> & lt; mi> r< / mi> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mrow> & lt;莫类= " MathClass-open "祝辞{& lt; / mo> & lt; mrow> & lt; msub> & lt; mi> M< / mi> & lt; mrow> & lt; mi> s< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; / mrow> & lt; / msub> & lt; mo>, & lt; / mo> & lt; mi> F< / mi> & lt; / mrow> & lt;莫类=“MathClass-close”祝辞}& lt; / mo> & lt; / mrow> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>当F&lt； 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></mrow></ mrow></mrow></ mrow></math&gt；χ (F)≥max {R + 1，4} <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0017”威利:位置= "方程/ jgt23210 -数学- 0017. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> \ unicode {x003C7} & lt; / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mi&gt F< / mi> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; mo> \ unicode {x02265} & lt; / mo> & lt; mi> max< / mi> & lt; mrow> & lt;莫类=“MathClass-open祝辞{& lt; / mo> & lt; mrow> & lt; mi> r< / mi> & lt; mo> \ unicode {x0002B} & lt; / mo> & lt; mn> 1 & lt; / mn> & lt; mo>, & lt; / mo> & lt; mn> 4 & lt; / mn> & lt; / mrow> & lt;莫类= " MathClass-close "祝辞}& lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>．

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Extremal Problems for a Matching and Any Other Graph

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 .