Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Given a simple graph $G=(V,E)$, a subset of $E$ is called a triangle cover if it intersects each triangle of $G$. Let $\nut(G)$ and $\taut(G)$ denote the maximum number of pairwise edge-disjoint triangles in $G$ and the minimum cardinality of a triangle cover of $G$, respectively. Tuza conjectured in 1981 that $\taut(G)/\nut(G)\le2$ holds for every graph $G$. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr TG$ of triangles covers all edges in $G$. We show that a triangle cover of $G$ with cardinality at most $2\nut(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $\nut(G)/|\mathscr TG|\ge\frac13$, (ii) $\nut(G)/|E|\ge\frac14$, (iii) $|E|/|\mathscr TG|\ge2$. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithms