A generalization of Tuza's conjecture

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.