Extremal Problems for a Matching and Any Other Graph

IF 0.9 3区 数学 Q2 MATHEMATICS
Xiutao Zhu, Yaojun Chen
{"title":"Extremal Problems for a Matching and Any Other Graph","authors":"Xiutao Zhu,&nbsp;Yaojun Chen","doi":"10.1002/jgt.23210","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>For a family of graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> &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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\"&gt;\\unicode{x02131}&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math>, a graph is called <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\"&gt;\\unicode{x02131}&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math>-free if it does not contain any member of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\"&gt;\\unicode{x02131}&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> as a subgraph. The generalized Turán number <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\"&gt;\\unicode{x02131}&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> is the maximum number of <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> in an <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math>-vertex <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ℱ</mi>\n </mrow>\n </mrow>\n <annotation> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\"&gt;\\unicode{x02131}&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math>-free graph and <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\"&gt;\\unicode{x02131}&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\"&gt;\\unicode{x02131}&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math>, that is, the classical Turán number. Let <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;\\unicode{x0002B}&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> be a matching on <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;\\unicode{x0002B}&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> edges and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> be any graph. In this paper, we determine <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo class=\"MathClass-open\"&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;\\unicode{x0002B}&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mo class=\"MathClass-close\"&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</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 <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo class=\"MathClass-open\"&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;\\unicode{x0002B}&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mo class=\"MathClass-close\"&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> for <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> being any non-bipartite graph or some bipartite graphs. Furthermore, we determine <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mtext&gt;ex&lt;/mtext&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mo class=\"MathClass-open\"&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;\\unicode{x0002B}&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mo class=\"MathClass-close\"&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> when <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> &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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math> is color critical with <span></span><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> &lt;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\"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;\\unicode{x003C7}&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;\\unicode{x02265}&lt;/mo&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;mrow&gt;&lt;mo class=\"MathClass-open\"&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;\\unicode{x0002B}&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;mo class=\"MathClass-close\"&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;</annotation>\n </semantics></math>.</p>\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 <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> , a graph is called <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> -free if it does not contain any member of <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> as a subgraph. The generalized Turán number ex ( n , K r , ) <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> is the maximum number of K r <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> in an n <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> -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.png"><mrow><mrow><mi class="MJX-tex-caligraphic" mathvariant="normal">\unicode{x02131}</mi></mrow></mrow></math> -free graph and ex ( n , K 2 , ) = ex ( n , ) <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> , that is, the classical Turán number. Let M s + 1 <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> be a matching on s + 1 <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> edges 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><mi>F</mi></mrow></mrow></math> be any graph. In this paper, we determine 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" 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> 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 } ) <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> 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><mi>F</mi></mrow></mrow></math> being any non-bipartite graph or some bipartite graphs. Furthermore, we determine 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" 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> when F <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> is color critical with 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" 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> .

匹配图和其他图的极值问题
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"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi class=" mjx - text - igrapigrapic " mathvariant="normal"&gt;\unicode{x02131}&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/math&gt;,一个图形被称为<s:1> &lt;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"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi class=" mjx - text - igrapigrapic " mathvariant="normal"&gt;\unicode{x02131}&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/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"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi class=" mjx - text - igrapigrapic " mathvariant="normal"&gt;\unicode{x02131}&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/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&gt; & lt; mrow&gt; & lt; mtext&gt; ex&lt; / mtext&gt; & lt; mrow&gt; & lt; mo&gt; (& lt; / mo&gt; & lt; mrow&gt; & lt; mi&gt; n&lt; / mi&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; msub&gt; & lt; mi&gt; K&lt; / mi&gt; & lt; mi&gt; r&lt; / mi&gt; & lt; / msub&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mi类=“MJX-tex-caligraphic”mathvariant =“正常”祝辞\ unicode {x02131} & lt; / mi&gt; & lt; / mrow&gt; & lt; mo&gt;) & lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;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&gt; & lt; mrow&gt; & lt; msub&gt; & lt; mi&gt; K&lt; / mi&gt; & lt; mi&gt; r&lt; / mi&gt; & lt; / msub&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;in an &lt;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"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt; /mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/math&gt;-vertex _ (&lt;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&gt;&lt;mrow&gt;&lt;mi class=" mjx - text - calligraphy " mathvariant="normal"&gt;\unicode{x02131}&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/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&gt; & lt; mrow&gt; & lt; mtext&gt; ex&lt; / mtext&gt; & lt; mrow&gt; & lt; mo&gt; (& lt; / mo&gt; & lt; mrow&gt; & lt; mi&gt; n&lt; / mi&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; msub&gt; & lt; mi&gt; K&lt; / mi&gt; & lt; mn&gt; 2 & lt; / mn&gt; & lt; / msub&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mi类=“MJX-tex-caligraphic”mathvariant = "正常"祝辞\ unicode {x02131} & lt; / mi&gt; & lt; / mrow&gt; & lt; mo&gt;) & lt; / mo&gt; & lt; / mrow&gt; & lt; mo&gt; = & lt; / mo&gt; & lt; mtext&gt; ex&lt; / mtext&gt; & lt; mrow&gt; & lt; mo&gt; (& lt; / mo&gt; & lt; mrow&gt; & lt; mi&gt; n&lt; / mi&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mi类=“MJX-tex-caligraphic”mathvariant =“正常”祝辞\ unicode {x02131} & lt; / mi&gt; & lt; / mrow&gt; & lt; mo&gt;) & lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;,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&gt; & lt; mrow&gt; & lt; msub&gt; & lt; mi&gt; M&lt; / mi&gt; & lt; mrow&gt; & lt; mi&gt; s&lt; / mi&gt; & lt; mo&gt; \ unicode {x0002B} & lt; / mo&gt; & lt; mn&gt; 1 & lt; / mn&gt; & lt; / mrow&gt; & lt; / msub&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;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&gt; & lt; mrow&gt; & lt; mi&gt; s&lt; / mi&gt; & lt; mo&gt; \ unicode {x0002B} & lt; / mo&gt; & lt; mn&gt; 1 & lt; / mn&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;edge and F&lt; 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"&gt;&lt;mrow&gt;&lt; /mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;是任意图形。 在本文中,我们确定ex (n),K r,{M s + 1,F}) &lt;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&gt; & lt; mrow&gt; & lt; mtext&gt; ex&lt; / mtext&gt; & lt; mrow&gt; & lt; mo&gt; (& lt; / mo&gt; & lt; mrow&gt; & lt; mi&gt; n&lt; / mi&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; msub&gt; & lt; mi&gt; K&lt; / mi&gt; & lt; mi&gt; r&lt; / mi&gt; & lt; / msub&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mrow&gt; & lt;莫类= " MathClass-open "祝辞{& lt; / mo&gt; & lt; mrow&gt; & lt; msub&gt; & lt; mi&gt; M&lt; / mi&gt; & lt; mrow&gt; & lt; mi&gt; s&lt; / mi&gt; & lt; mo&gt; \ unicode {x0002B} & lt; / mo&gt; & lt; mn&gt; 1 & lt; / mn&gt; & lt; / mrow&gt; & lt; / msub&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mi&gt; F&lt; / mi&gt; & lt; / mrow&gt; & lt;莫类=“MathClass-close”祝辞}& lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; mo&gt;) & lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;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}) &lt;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&gt; & lt; mrow&gt; & lt; mtext&gt; ex&lt; / mtext&gt; & lt; mrow&gt; & lt; mo&gt; (& lt; / mo&gt; & lt; mrow&gt; & lt; mi&gt; n&lt; / mi&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mrow&gt; & lt;莫类=“MathClass-open祝辞{& lt; / mo&gt; & lt; mrow&gt; & lt; msub&gt; & lt; mi&gt; M&lt; / mi&gt; & lt; mrow&gt; & lt; mi&gt; s&lt; / mi&gt; & lt; mo&gt; \ unicode {x0002B} & lt; / mo&gt; & lt; mn&gt; 1 & lt; / mn&gt; & lt; / mrow&gt; & lt; / msub&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mi&gt; F&lt; / mi&gt; & lt; / mrow&gt; & lt;莫类= " MathClass-close "祝辞}& lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; mo&gt;) & lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;for F&lt; 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"&gt;&lt;mrow&gt;&lt;mrow&gt;&lt; /mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/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}) &lt;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&gt; & lt; mrow&gt; & lt; mtext&gt; ex&lt; / mtext&gt; & lt; mrow&gt; & lt; mo&gt; (& lt; / mo&gt; & lt; mrow&gt; & lt; mi&gt; n&lt; / mi&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; msub&gt; & lt; mi&gt; K&lt; / mi&gt; & lt; mi&gt; r&lt; / mi&gt; & lt; / msub&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mrow&gt; & lt;莫类= " MathClass-open "祝辞{& lt; / mo&gt; & lt; mrow&gt; & lt; msub&gt; & lt; mi&gt; M&lt; / mi&gt; & lt; mrow&gt; & lt; mi&gt; s&lt; / mi&gt; & lt; mo&gt; \ unicode {x0002B} & lt; / mo&gt; & lt; mn&gt; 1 & lt; / mn&gt; & lt; / mrow&gt; & lt; / msub&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mi&gt; F&lt; / mi&gt; & lt; / mrow&gt; & lt;莫类=“MathClass-close”祝辞}& lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; mo&gt;) & lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;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"&gt;&lt;mrow&gt;&lt; /mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/mrow&gt;&lt;/ mrow&gt;&lt;/math&gt;χ (F)≥max {R + 1,4} &lt;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&gt; & lt; mrow&gt; & lt; mi&gt; \ unicode {x003C7} & lt; / mi&gt; & lt; mrow&gt; & lt; mo&gt; (& lt; / mo&gt; & lt; mi&gt F&lt; / mi&gt; & lt; mo&gt;) & lt; / mo&gt; & lt; / mrow&gt; & lt; mo&gt; \ unicode {x02265} & lt; / mo&gt; & lt; mi&gt; max&lt; / mi&gt; & lt; mrow&gt; & lt;莫类=“MathClass-open祝辞{& lt; / mo&gt; & lt; mrow&gt; & lt; mi&gt; r&lt; / mi&gt; & lt; mo&gt; \ unicode {x0002B} & lt; / mo&gt; & lt; mn&gt; 1 & lt; / mn&gt; & lt; mo&gt;, & lt; / mo&gt; & lt; mn&gt; 4 & lt; / mn&gt; & lt; / mrow&gt; & lt;莫类= " MathClass-close "祝辞}& lt; / mo&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / mrow&gt; & lt; / math&gt;.
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来源期刊
Journal of Graph Theory
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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