An improved upper bound for Tuza's conjecture via 2-colorable triangle families

Tuza’s conjecture states that for any graph $G$, the minimum size of a triangle transversal $\tau (G)$ is at most twice the maximum size of a set of edge-disjoint triangles $\nu (G)$. In this note, we prove $\tau (G)\le \frac{63}{22}\nu (G)$, improving the previous bound $\tau (G)\le \frac{66}{23}\nu (G)$ established by Haxell in 1999. The key observation is that for a “2-colorable” family of triangles $\mathcal{F}$, where each triangle has two blue edges and one red edge, we can obtain $\tau (\mathcal{F})\le (1+\sqrt{3})\nu (\mathcal{F})$.