On the Generalized Turán Problem for Odd Cycles

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-09-10 DOI:10.1137/24m1632632

Csongor Beke, Oliver Janzer

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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.

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论奇数循环的广义图兰问题

SIAM 离散数学杂志》，第 38 卷第 3 期，第 2416-2428 页，2024 年 9 月。摘要1984 年，厄尔多斯猜想，[math]顶点上任何无三角形图中的五角星数至多为 [math]，这在五角星的平衡炸开中是很尖锐的。作为这一结果在更长循环上的扩展，我们证明，对于每个奇数 [math]，只要 [math] 足够大，[math] 的平衡炸开（唯一地）就能最大化[math]顶点上无[math]图中的[math]循环数。我们还证明，如果不假定 [math] 足够大，这一结果就不再成立。我们的结果加强了 Grzesik 和 Kielak 的结果，他们证明了对于每个奇数 [math]，[math] 的平衡吹大可以最大化具有给定顶点数且没有长度小于 [math] 的奇数循环的图中的 [math] 循环数。我们进一步证明，如果[math]和[math]都是奇数，且[math]与[math]相比足够大，那么[math]的平衡炸开并不能渐近地最大化[math]顶点上无[math]图中的[math]循环数。这推翻了格热西克和基拉克的猜想。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

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.