Generalized Tuza’s Conjecture for Random Hypergraphs

IF 0.9 3区 数学 Q2 MATHEMATICS
Abdul Basit, David Galvin
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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2260-2288, September 2024.
Abstract. A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an [math]-uniform hypergraph ([math]-graph) [math], let [math] be the minimum size of a cover of edges by [math]-sets of vertices, and let [math] be the maximum size of a set of edges pairwise intersecting in fewer than [math] vertices. Aharoni and Zerbib proposed the following generalization of Tuza’s conjecture: For any [math]-graph [math], [math]. Let [math] be the uniformly random [math]-graph on [math] vertices. We show that for [math] and any [math], [math] satisfies the Aharoni–Zerbib conjecture with high probability (w.h.p.), i.e., with probability approaching 1 as [math]. We also show that there is a [math] such that for any [math] and any [math], [math] w.h.p. Furthermore, we may take [math], for any [math], by restricting to sufficiently large [math] (depending on [math]).
随机超图的广义图扎猜想
SIAM 离散数学杂志》,第 38 卷第 3 期,第 2260-2288 页,2024 年 9 月。 摘要图扎(Tuza)的一个著名猜想指出,在任何有限图中,边覆盖三角形的最小尺寸最多是边相交三角形集合最大尺寸的两倍。对于一个[math]-均匀超图([math]-图)[math],设[math]是[math]-顶点集的边覆盖的最小尺寸,设[math]是少于[math]个顶点的边成对相交的边集的最大尺寸。阿哈罗尼和泽尔毕布对图扎猜想提出了如下概括:对于任意[math]图[math],[math]。让 [math] 成为[math]顶点上的均匀随机[math]图。我们证明,对于[math]和任意[math],[math]以高概率(w.h.p.)满足阿哈龙-泽尔毕布猜想,即随着[math]的增大,概率趋近于 1。我们还证明存在一个 [math],使得对于任意 [math] 和任意 [math],[math] w.h.p. 而且,对于任意 [math],我们可以通过限制到足够大的 [math](取决于 [math])来取 [math]。
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来源期刊
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.
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