{"title":"The burning number conjecture holds asymptotically","authors":"Sergey Norin, Jérémie Turcotte","doi":"10.1016/j.jctb.2024.05.003","DOIUrl":null,"url":null,"abstract":"<div><p>The burning number <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mrow><mo>⌈</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>⌉</mo></mrow></math></span> for all connected graphs <em>G</em> on <em>n</em> vertices. We prove that this conjecture holds asymptotically, that is <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span>.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-05-29","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/S009589562400042X","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 burning number of a graph G is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that for all connected graphs G on n vertices. We prove that this conjecture holds asymptotically, that is .
图 G 的燃烧数 b(G)是指如果每转一圈都有新的火开始燃烧，并且已有的火蔓延到所有相邻的顶点，则烧毁图中所有顶点所需的最小圈数。Bonato 等人（2016 年）提出的燃烧次数猜想假设，对于 n 个顶点上的所有连通图 G，b(G)≤⌈n⌉。我们证明这一猜想近似成立，即 b(G)≤(1+o(1))n。
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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