Positive Co-Degree Turán Number for C5 and C5−

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-12-30 DOI:10.1002/jgt.23206

Zhuo Wu

{"title":"Positive Co-Degree Turán Number for C5 and C5−","authors":"Zhuo Wu","doi":"10.1002/jgt.23206","DOIUrl":null,"url":null,"abstract":"The minimum positive co-degree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msubsup>\n <mi>δ</mi>\n \n <mrow>\n <mi>r</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>+</mo>\n </msubsup>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</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:jgt23206:jgt23206-math-0001\" wiley:location=\"equation/jgt23206-math-0001.png\"><mrow><mrow><msubsup><mi>\\unicode{x003B4}</mi><mrow><mi>r</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mo>\\unicode{x0002B}</mo></msubsup><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> of a nonempty <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0002\" wiley:location=\"equation/jgt23206-math-0002.png\"><mrow><mrow><mi>r</mi></mrow></mrow></math></annotation>\n </semantics></math>-graph <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:jgt23206:jgt23206-math-0003\" wiley:location=\"equation/jgt23206-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> is the maximum <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0004\" wiley:location=\"equation/jgt23206-math-0004.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math> such that if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>S</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0005\" wiley:location=\"equation/jgt23206-math-0005.png\"><mrow><mrow><mi>S</mi></mrow></mrow></math></annotation>\n </semantics></math> is an <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>r</mi>\n \n <mo>−</mo>\n \n <mn>1</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:jgt23206:jgt23206-math-0006\" wiley:location=\"equation/jgt23206-math-0006.png\"><mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>-set contained in a hyperedge of <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:jgt23206:jgt23206-math-0007\" wiley:location=\"equation/jgt23206-math-0007.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>S</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0008\" wiley:location=\"equation/jgt23206-math-0008.png\"><mrow><mrow><mi>S</mi></mrow></mrow></math></annotation>\n </semantics></math> is contained in at least <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0009\" wiley:location=\"equation/jgt23206-math-0009.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math> hyperedges of <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:jgt23206:jgt23206-math-0010\" wiley:location=\"equation/jgt23206-math-0010.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>. For any <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0011\" wiley:location=\"equation/jgt23206-math-0011.png\"><mrow><mrow><mi>r</mi></mrow></mrow></math></annotation>\n </semantics></math>-graph <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:jgt23206:jgt23206-math-0012\" wiley:location=\"equation/jgt23206-math-0012.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math>, the positive degree Turán number <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mtext>co</mtext>\n \n <mo>+</mo>\n </msup>\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:jgt23206:jgt23206-math-0013\" wiley:location=\"equation/jgt23206-math-0013.png\"><mrow><mrow><msup><mtext>co</mtext><mo>\\unicode{x0002B}</mo></msup><mtext>ex</mtext><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is defined as the maximum value of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msubsup>\n <mi>δ</mi>\n \n <mrow>\n <mi>r</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>+</mo>\n </msubsup>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</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:jgt23206:jgt23206-math-0014\" wiley:location=\"equation/jgt23206-math-0014.png\"><mrow><mrow><msubsup><mi>\\unicode{x003B4}</mi><mrow><mi>r</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow><mo>\\unicode{x0002B}</mo></msubsup><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> over all <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:jgt23206:jgt23206-math-0015\" wiley:location=\"equation/jgt23206-math-0015.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:jgt23206:jgt23206-math-0016\" wiley:location=\"equation/jgt23206-math-0016.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math>-free nonempty <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0017\" wiley:location=\"equation/jgt23206-math-0017.png\"><mrow><mrow><mi>r</mi></mrow></mrow></math></annotation>\n </semantics></math>-graphs <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:jgt23206:jgt23206-math-0018\" wiley:location=\"equation/jgt23206-math-0018.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>. In this paper, we determine the positive degree Turán number for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mn>5</mn>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0019\" wiley:location=\"equation/jgt23206-math-0019.png\"><mrow><mrow><msub><mi>C</mi><mn>5</mn></msub></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msubsup>\n <mi>C</mi>\n \n <mn>5</mn>\n \n <mo>−</mo>\n </msubsup>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0020\" wiley:location=\"equation/jgt23206-math-0020.png\"><mrow><mrow><msubsup><mi>C</mi><mn>5</mn><mo>\\unicode{x02212}</mo></msubsup></mrow></mrow></math></annotation>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"25-30"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23206","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23206","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

The minimum positive co-degree $δ_{r - 1}^{+} (H)$ of a nonempty $r$ -graph $H$ is the maximum $k$ such that if $S$ is an $(r - 1)$ -set contained in a hyperedge of $H$ , then $S$ is contained in at least $k$ hyperedges of $H$ . For any $r$ -graph $F$ , the positive degree Turán number ${co}^{+} ex (n, F)$ is defined as the maximum value of $δ_{r - 1}^{+} (H)$ over all $n$ -vertex $F$ -free nonempty $r$ -graphs $H$ . In this paper, we determine the positive degree Turán number for $C_{5}$ and $C_{5}^{-}$ .

查看原文本刊更多论文

求助全文

约1分钟内获得全文求助全文

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 .