{"title":"在两个连续偶数长度的周期上","authors":"Jun Gao, Binlong Li, Jie Ma, Tianying Xie","doi":"10.1002/jgt.23074","DOIUrl":null,"url":null,"abstract":"<p>Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two. We prove the following average degree counterpart that every <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-vertex graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with at least <span></span><math>\n <semantics>\n <mrow>\n <mfrac>\n <mn>5</mn>\n \n <mn>2</mn>\n </mfrac>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\frac{5}{2}(n-1)$</annotation>\n </semantics></math> edges, unless <span></span><math>\n <semantics>\n <mrow>\n <mn>4</mn>\n \n <mo>|</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $4|(n-1)$</annotation>\n </semantics></math> and every block of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is a clique <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n \n <mn>5</mn>\n </msub>\n </mrow>\n <annotation> ${K}_{5}$</annotation>\n </semantics></math>, contains two cycles of consecutive even lengths. Our proof is mainly based on structural analysis, and a crucial step which may be of independent interest shows that the same conclusion holds for every 3-connected graph with at least six vertices. This solves a special case of a conjecture of Verstraëte. The quantitative bound is tight and also provides the optimal extremal number for cycles of length two modulo four.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 2","pages":"225-238"},"PeriodicalIF":0.9000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On two cycles of consecutive even lengths\",\"authors\":\"Jun Gao, Binlong Li, Jie Ma, Tianying Xie\",\"doi\":\"10.1002/jgt.23074\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two. We prove the following average degree counterpart that every <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math>-vertex graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> with at least <span></span><math>\\n <semantics>\\n <mrow>\\n <mfrac>\\n <mn>5</mn>\\n \\n <mn>2</mn>\\n </mfrac>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\frac{5}{2}(n-1)$</annotation>\\n </semantics></math> edges, unless <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>4</mn>\\n \\n <mo>|</mo>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $4|(n-1)$</annotation>\\n </semantics></math> and every block of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> is a clique <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>K</mi>\\n \\n <mn>5</mn>\\n </msub>\\n </mrow>\\n <annotation> ${K}_{5}$</annotation>\\n </semantics></math>, contains two cycles of consecutive even lengths. Our proof is mainly based on structural analysis, and a crucial step which may be of independent interest shows that the same conclusion holds for every 3-connected graph with at least six vertices. This solves a special case of a conjecture of Verstraëte. The quantitative bound is tight and also provides the optimal extremal number for cycles of length two modulo four.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"106 2\",\"pages\":\"225-238\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-01-24\",\"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.23074\",\"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.23074","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two. We prove the following average degree counterpart that every -vertex graph with at least edges, unless and every block of is a clique , contains two cycles of consecutive even lengths. Our proof is mainly based on structural analysis, and a crucial step which may be of independent interest shows that the same conclusion holds for every 3-connected graph with at least six vertices. This solves a special case of a conjecture of Verstraëte. The quantitative bound is tight and also provides the optimal extremal number for cycles of length two modulo four.
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 .