New bounds on a generalization of Tuza's conjecture

TL;DR

This paper investigates a generalized form of Tuza's conjecture for hypergraphs, proving it for cases where the maximum independent set size is at most three, and explores related bounds and fractional variants.

Contribution

It proves the generalized Tuza's conjecture for hypergraphs with maximum independent set size up to three and derives bounds for other parameters and fractional versions.

Findings

The conjecture holds when ^{(k-1)}(H)  3.

Established bounds on ^{(m)}(H)/^{(m)}(H) for various m.

Derived bounds on fractional analogues of these parameters.

Abstract

For a $k$-uniform hypergraph $H$, let $\nu^{(m)}(H)$ denote the maximum size of a set $S$ of edges of $H$ whose pairwise intersection has size less than $m$. Let $\tau^{(m)}(H)$ denote the minimum size of a set $S$ of $m$-sets of $V(H)$ such that every edge of $H$ contains some $m$-set from $S$. A conjecture by Aharoni and Zerbib, which generalizes a conjecture of Tuza on the size of minimum edge covers of triangles of a graph, states that for a $k$-uniform hypergraph $H$, $\tau^{(k - 1)}(H)/\nu^{(k - 1)}(H) \leq \left \lceil \frac{k + 1}{2} \right \rceil$. In this paper, we show that this generalization of Tuza's conjecture holds when $\nu^{(k - 1)}(H) \leq 3$. As a corollary, we obtain a graph class which satisfies Tuza's conjecture. We also prove various bounds on $\tau^{(m)}(H)/\nu^{(m)}(H)$ for other values of $m$ as well as some bounds on the fractional analogues of these numbers.