{"title":"Dirac-type theorems for long Berge cycles in hypergraphs","authors":"Alexandr Kostochka , Ruth Luo , Grace McCourt","doi":"10.1016/j.jctb.2024.05.001","DOIUrl":null,"url":null,"abstract":"<div><p>The famous Dirac's Theorem gives an exact bound on the minimum degree of an <em>n</em>-vertex graph guaranteeing the existence of a hamiltonian cycle. In the same paper, Dirac also observed that a graph with minimum degree at least <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> contains a cycle of length at least <span><math><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least <em>k</em> in <em>r</em>-uniform, <em>n</em>-vertex hypergraphs for all combinations of <span><math><mi>k</mi><mo>,</mo><mi>r</mi></math></span> and <em>n</em> with <span><math><mn>3</mn><mo>≤</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>. The bounds differ for different ranges of <em>r</em> compared to <em>n</em> and <em>k</em>.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895624000406","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The famous Dirac's Theorem gives an exact bound on the minimum degree of an n-vertex graph guaranteeing the existence of a hamiltonian cycle. In the same paper, Dirac also observed that a graph with minimum degree at least contains a cycle of length at least . The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least k in r-uniform, n-vertex hypergraphs for all combinations of and n with . The bounds differ for different ranges of r compared to n and k.
著名的狄拉克定理给出了 n 个顶点图的最小度的精确约束，保证了哈密顿循环的存在。在同一篇文章中，狄拉克还观察到一个最小度至少为 k≥2 的图包含一个长度至少为 k+1 的循环。本文的目的有两个：我们证明了类似类型的哈密顿贝格循环以及长度至少为 k 的 r-uniform n 顶点超图中的贝格循环的精确边界，适用于 3≤r,k≤n 的 k、r 和 n 的所有组合。与 n 和 k 相比，r 的范围不同，界限也不同。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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