Generalized Tuza's conjecture for random hypergraphs

TL;DR

This paper proves that a generalized version of Tuza's conjecture holds with high probability for random hypergraphs with small uniformity, and provides bounds for larger uniformities, advancing understanding of hypergraph covering and matching properties.

Contribution

It establishes probabilistic validation of a generalized Tuza's conjecture for random hypergraphs with uniformity 3 to 5, and provides bounds for higher uniformities.

Findings

For r=3,4,5, the conjecture holds with high probability.

For r ≥ 6, the ratio is bounded by a constant less than 1, with high probability.

The bounds improve to below 1/2 + ε for large r.

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 $r$-uniform hypergraph ($r$-graph) $G$, let $\tau(G)$ be the minimum size of a cover of edges by $(r-1)$-sets of vertices, and let $\nu(G)$ be the maximum size of a set of edges pairwise intersecting in fewer than $r-1$ vertices. Aharoni and Zerbib proposed the following generalization of Tuza's conjecture: $$ \text{For any $r$-graph $G$, $\tau(G)/\nu(G) \leq \lceil(r+1)/2\rceil$.} $$ Let $H_r(n,p)$ be the uniformly random $r$-graph on $n$ vertices. We show that, for $r \in \{3, 4, 5\}$ and any $p = p(n)$, $H_r(n,p)$ satisfies the Aharoni-Zerbib conjecture with high probability (i.e., with probability approaching 1 as $n \rightarrow \infty$). We also show that there is a $C < 1$ such that, for any $r \geq 6$ and any $p = p(n)$, $\tau(H_r(n, p))/\nu(H_r(n, p)) \leq C r$ with high probability. Furthermore, we may take $C < 1/2 + \varepsilon$, for any $\varepsilon > 0$, by restricting to sufficiently large $r$ (depending on $\varepsilon$).