Packing and covering directed triangles asymptotically

TL;DR

This paper improves bounds on the minimum number of arcs needed to intersect all directed triangles in a graph with at most t pairwise arc-disjoint triangles, advancing understanding of directed triangle packing and covering.

Contribution

It establishes an upper bound of 1.8t+o(n^2) arcs for covering all directed triangles, improving previous bounds, and provides constructions showing the bound's near-optimality.

Findings

Upper bound of 1.8t+o(n^2) arcs for covering directed triangles.

Existence of graphs with minimum covering sets of 1.5t-o(n^2) arcs.

Progress towards optimal bounds in directed triangle covering.

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.