Packing and covering directed triangles

TL;DR

This paper proves that in directed multigraphs with at most t disjoint directed triangles, a set of fewer than 2t arcs can intersect all triangles, confirming a 1990 conjecture by Tuza.

Contribution

It establishes an upper bound on the size of an arc set intersecting all directed triangles in terms of the maximum number of disjoint triangles, solving a longstanding open problem.

Findings

For t=0, the bound is trivial.

The bound is less than 2t arcs for graphs with t disjoint triangles.

The result confirms Tuza's 1990 conjecture for directed graphs.

Abstract

We prove that if a directed multigraph $D$ has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of less than $2t$ arcs in $D$ which meets all directed triangles in $D$, except in the trivial case $t=0$. This answers affirmatively a question of Tuza from 1990.