Generalized Tuza’s Conjecture for Random Hypergraphs

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]).