Internally 4-connected binary matroids with every element in three triangles (Q2338614)

The purpose of this paper is to prove that a binary internally 4-connected matroid \(M\) in which every element is in exactly three triangles has at least four elements \(e\) such that the simplification of \(M/e\), si\((M/e)\), is internally 4-connected. This theorem is first proved for matroids on at most thirteen elements. Then a sequence of lemmas is proved dealing with small cocircuits. The proof of the theorem follows then easily and elegantly. Moreover, an examle of a matroid \(M\) satisfying the hypothesis of the theorem is constructed in which si\((M/e)\) is not internally 4-connected for every \(e\).