A generalization of Tuza's conjecture

TL;DR

This paper extends Tuza's conjecture from graphs to hypergraphs, proposing a broader framework for covering and packing edges, and explores the conjecture's implications, known results, and fractional variants.

Contribution

It introduces a generalized setting for Tuza's conjecture in hypergraphs, proposes a new conjecture on the ratio of covering to packing numbers, and extends known results to this broader context.

Findings

Most known results on Tuza's conjecture extend to the hypergraph setting.

Proves some general bounds on the ratio of covering to packing numbers.

Studies fractional and k-partite hypergraph cases.

Abstract

A famous conjecture of Tuza \cite{tuza} is that the minimal number of edges needed to cover all triangles in a graph is at most twice the maximal number of edge-disjoint triangles. We propose a wider setting for this conjecture. For a hypergraph $H$ let $\nu^{(m)}(H)$ be the maximal size of a collection of edges, no two of which share $m$ or more vertices, and let $\tau^{(m)}(H)$ be the minimal size of a collection $C$ of sets of $m$ vertices, such that every edge in $H$ contains a set from $C$. We conjecture that the maximal ratio $\tau^{(m)}(H)/\nu^{(m)}(H)$ is attained in hypergraphs for which $\nu^{(m)}(H)=1$. This would imply, in particular, the following generalization of Tuza's conjecture: if $H$ is $3$-uniform, then $\tau^{(2)}(H)/\nu^{(2)}(H) \le 2$. (Tuza's conjecture is the case in which $H$ is the set of all triples of vertices of triangles in the graph). We show that most known results on Tuza's conjecture go over to this more general setting. We also prove some general results on the ratio $\tau^{(m)}(H)/\nu^{(m)}(H)$, and study the fractional versions and the case of $k$-partite hypergraphs.