Weighted exchange distance of basis pairs

Two pairs of disjoint bases P1 = (R1, B1) and P2 = (R2, B2) of a matroid M are called equivalent if P1 can be transformed into P2 by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever R1 U B1 = R2 U B2. A strengthening of the conjecture was proposed by Hamidoune, stating that the minimum length of an exchange is at most the rank of the matroid. We propose a weighted variant of Hamidoune's conjecture, where the weight of an exchange depends on the weights of the exchanged elements. We prove the conjecture for several matroid classes: strongly base orderable matroids, split matroids, graphic matroids of wheels, and spikes. (c) 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).