{"title":"On the Generalized Turán Problem for Odd Cycles","authors":"Csongor Beke, Oliver Janzer","doi":"10.1137/24m1632632","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2416-2428, September 2024. <br/> Abstract. In 1984, Erdős conjectured that the number of pentagons in any triangle-free graph on [math] vertices is at most [math], which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami et al. As an extension of this result for longer cycles, we prove that for each odd [math], the balanced blow-up of [math] (uniquely) maximizes the number of [math]-cycles among [math]-free graphs on [math] vertices, as long as [math] is sufficiently large. We also show that this is no longer true if [math] is not assumed to be sufficiently large. Our result strengthens results of Grzesik and Kielak who proved that for each odd [math], the balanced blow-up of [math] maximizes the number of [math]-cycles among graphs with a given number of vertices and no odd cycles of length less than [math]. We further show that if [math] and [math] are odd and [math] is sufficiently large compared to [math], then the balanced blow-up of [math] does not asymptotically maximize the number of [math]-cycles among [math]-free graphs on [math] vertices. This disproves a conjecture of Grzesik and Kielak.","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":"53 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/24m1632632","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
SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2416-2428, September 2024. Abstract. In 1984, Erdős conjectured that the number of pentagons in any triangle-free graph on [math] vertices is at most [math], which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami et al. As an extension of this result for longer cycles, we prove that for each odd [math], the balanced blow-up of [math] (uniquely) maximizes the number of [math]-cycles among [math]-free graphs on [math] vertices, as long as [math] is sufficiently large. We also show that this is no longer true if [math] is not assumed to be sufficiently large. Our result strengthens results of Grzesik and Kielak who proved that for each odd [math], the balanced blow-up of [math] maximizes the number of [math]-cycles among graphs with a given number of vertices and no odd cycles of length less than [math]. We further show that if [math] and [math] are odd and [math] is sufficiently large compared to [math], then the balanced blow-up of [math] does not asymptotically maximize the number of [math]-cycles among [math]-free graphs on [math] vertices. This disproves a conjecture of Grzesik and Kielak.
期刊介绍:
SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution.
Topics include but are not limited to:
properties of and extremal problems for discrete structures
combinatorial optimization, including approximation algorithms
algebraic and enumerative combinatorics
coding and information theory
additive, analytic combinatorics and number theory
combinatorial matrix theory and spectral graph theory
design and analysis of algorithms for discrete structures
discrete problems in computational complexity
discrete and computational geometry
discrete methods in computational biology, and bioinformatics
probabilistic methods and randomized algorithms.