{"title":"A localized approach for Turán number of long cycles","authors":"Kai Zhao, Xiao-Dong Zhang","doi":"10.1002/jgt.23191","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><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:jgt23191:jgt23191-math-0001\" wiley:location=\"equation/jgt23191-math-0001.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> be a simple graph and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>c</mi>\n \n <mi>G</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>e</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:jgt23191:jgt23191-math-0002\" wiley:location=\"equation/jgt23191-math-0002.png\"><mrow><mrow><msub><mi>c</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> denote the length of the longest cycle containing an edge <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>e</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23191:jgt23191-math-0003\" wiley:location=\"equation/jgt23191-math-0003.png\"><mrow><mrow><mi>e</mi></mrow></mrow></math></annotation>\n </semantics></math> if there exists a cycle containing <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>e</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23191:jgt23191-math-0004\" wiley:location=\"equation/jgt23191-math-0004.png\"><mrow><mrow><mi>e</mi></mrow></mrow></math></annotation>\n </semantics></math>, and 2 otherwise. We prove that the summation of 2 divided by <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>c</mi>\n \n <mi>G</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>e</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:jgt23191:jgt23191-math-0005\" wiley:location=\"equation/jgt23191-math-0005.png\"><mrow><mrow><msub><mi>c</mi><mi>G</mi></msub><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> over all edges in an <span></span><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:jgt23191:jgt23191-math-0006\" wiley:location=\"equation/jgt23191-math-0006.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math>-vertex graph <span></span><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:jgt23191:jgt23191-math-0007\" wiley:location=\"equation/jgt23191-math-0007.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\n </semantics></math> is at most <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</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:jgt23191:jgt23191-math-0008\" wiley:location=\"equation/jgt23191-math-0008.png\"><mrow><mrow><mi>n</mi><mo>\\unicode{x02212}</mo><mn>1</mn></mrow></mrow></math></annotation>\n </semantics></math>, and characterize all <span></span><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:jgt23191:jgt23191-math-0009\" wiley:location=\"equation/jgt23191-math-0009.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math>-vertex extremal graphs that achieve the bound, thereby extending the classic Erdős–Gallai theorem on cycles.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"582-607"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-08","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.23191","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a simple graph and denote the length of the longest cycle containing an edge if there exists a cycle containing , and 2 otherwise. We prove that the summation of 2 divided by over all edges in an -vertex graph is at most , and characterize all -vertex extremal graphs that achieve the bound, thereby extending the classic Erdős–Gallai theorem on cycles.
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.
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