A Bound Below 2.8 for Tuza's Conjecture

Let $\nu (G)$ be the maximum number of edge-disjoint triangles in a graph $G$ and $\tau (G)$ the minimum number of edges meeting every triangle. Tuza conjectured that $\tau (G)\le 2\nu (G)$. We prove that $\tau (G)\le (165\mathrm{/}59)\nu (G)$. The constant $165\mathrm{/}59\approx 2.797$ improves the bound $66\mathrm{/}23\approx 2.870$ that Haxell proved in 1999. The key observation is that, for a suitable red-blue coloring, the families left over in Haxell’s construction contain every triangle with exactly one red edge. Such a family $\mathcal{F}$ admits an exchange that forces certain red edges to lie in a single triangle once the blue edges of a maximum packing are deleted, which gives $\tau (\mathcal{F})\le (8\mathrm{/}3)\nu (\mathcal{F})$.