On matroids with many common bases (Q1301664)

Let \({\mathcal B}(M)\) denote the collection of bases of a matroid \(M.\) The author shows that if \(M_1\) and \(M_2\) are connected matroids having the same ground set and the symmetric difference \( {\mathcal B}(M_1)\Delta {\mathcal B}(M_2)\) has cardinality two, then, apart from a trivial exception, \(M_1\) and \(M_2\) are related via the relaxation operation. At the end of the paper the author formulates a conjecture that generalizes his result and the earlier one of \textit{K. Trümper} [J. Comb. Theory, Ser. B 32, 122-139 (1982; Zbl 0465.05022)].