Packing and covering directed triangles asymptotically

Abstract

A well-known conjecture of Tuza asserts that if a graph has at most $t$ pairwise edge-disjoint triangles, then it can be made triangle-free by removing at most $2t$ edges. If true, the factor 2 would be best possible. In the directed setting, also asked by Tuza, the analogous statement has recently been proven, however, the factor 2 is not optimal. In this paper, we show that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of at most $1.8t+o(n^2)$ arcs that meets all directed triangles. We complement our result by presenting two constructions of large directed graphs with $t\in \Omega (n^2)$ whose smallest such set has $1.5t-o(n^2)$ arcs.