In our recent paper titled Reconfiguration of basis pairs in regular matroids, we verify conjectures of White and of Gabow for the class of regular matroids. White’s conjecture aims at characterizing two basis sequences being reachable from each other by symmetric exchanges, which received a significant interest also in algebra due to its connection to toric ideals and Gröbner bases. Gabow’s conjecture suggests that matroids have the so-called serial symmetric exchange property.
Though finding a sequence of symmetric exchanges between basis sequences may seem to be a structural question purely on matroids, the problem is closely related to various topics such as fair allocations of indivisible goods, sampling common bases of two matroids, and the Carathéodory rank of matroid polytopes.