Tuza's conjecture for general graphs

Prove that for every simple graph G, the minimum size \(\tau(G)\) of a triangle transversal is at most twice the maximum size \(\nu(G)\) of a family of edge-disjoint triangles, that is, establish \(\tau(G)\leq 2\nu(G)\).

Background

Tuza's conjecture concerns the relationship between two parameters of a graph: the triangle-transversal number $\tau(G)$, the minimum number of edges whose deletion intersects every triangle, and the triangle-packing number $\nu(G)$, the maximum number of pairwise edge-disjoint triangles. The conjectured factor of 2 is known for several graph classes, including planar graphs, graphs with bounded treewidth, and threshold graphs, but the general case for arbitrary simple graphs remains unresolved. The paper improves the best previously known general upper bound from $66/23$ to $63/22$, without proving the conjectured factor of 2.