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(n2) arcs that meets all directed triangles. We complement our result by presenting two constructions of large directed graphs with t∈Ω(n2) whose smallest such set has 1.5t−o(n2) arcs.
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