Tuza's conjecture for binary geometries

TL;DR

This paper extends Tuza's conjecture from graphs to binary matroids, proving the conjecture for cographic matroids and exploring its validity in a broader combinatorial context.

Contribution

It generalizes Tuza's conjecture to binary matroids excluding the Fano plane and proves it for cographic matroids, advancing understanding in matroid theory.

Findings

Tuza's conjecture is valid for cographic matroids.

The conjecture is extended to binary matroids without Fano plane restrictions.

Provides new insights into the structure of binary matroids and triangle-free configurations.

Abstract

Tuza (A conjecture, in Proceedings of the Colloquia Mathematica Societatis Janos Bolyai, 1981) conjectured that $\tau(G) \le 2\nu(G)$ for all graphs $G$, where $\tau(G)$ is the minimum size of an edge set whose removal makes $G$ triangle-free, and $\nu(G)$ is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalise Tuza's conjecture to simple binary matroids that do not contain the Fano plane as a restriction. We prove that the geometric version of the conjecture holds for cographic matroids.