Weighted exchange distance of basis pairs

Abstract

Two pairs of disjoint bases ${\mathbf{P}}_1=(R_1,B_1)$ and ${\mathbf{P}}_2=(R_2,B_2)$ of a matroid $M$ are called equivalent if ${\mathbf{P}}_1$ can be transformed into ${\mathbf{P}}_2$ by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever $R_1\cup B_1=R_2\cup B_2$. A strengthening of the conjecture was proposed by Hamidoune, stating that 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.

20 pages, 5 figures