Tuza's Conjecture for Binary Geometries

Tuza [Finite and Infinite Sets, Proc. Colloq. Math. Soc. János Bolyai 37, North Holland, 1981, p. 888] conjectured that 𝜏⁡(𝐺)≤2⁢𝜈⁡(𝐺) for all graphs 𝐺, where 𝜏⁡(𝐺) is the minimum size of an edge set whose removal makes 𝐺 triangle-free and 𝜈⁡(𝐺) is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalize Tuza’s conjecture to simple binary matroids that do not contain the Fano plane as a restriction and prove that the geometric version of the conjecture holds for cographic matroids.