Packing and covering directed triangles

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.

4 pages, 1 figure. The previous version had an error in the argument for the strict inequality in the case where digons are permitted; this version fixes the error