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

Abstract

Tuza's conjecture states that for any graph $G$, the minimum size of a triangle transversal $τ(G)$ is at most twice the maximum size of a set of edge-disjoint triangles $ν(G)$. In this note, we prove $τ(G) \leq \frac{63}{22}ν(G)$, improving the previous bound $τ(G) \leq \frac{66}{23}ν(G)$ 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 $τ(\mathcal{F})\leq (1+\sqrt{3})ν(\mathcal{F})$.