Weighted exchange distance of basis pairs

TL;DR

This paper introduces a weighted exchange distance concept for basis pairs in matroids, proving a conjecture for specific matroid classes and extending understanding of basis exchange sequences.

Contribution

It proposes a weighted variant of Hamidoune's conjecture and proves it for several classes of matroids, advancing the theory of basis exchanges.

Findings

Proved the weighted exchange conjecture for strongly base orderable matroids

Extended the conjecture to split and graphic matroids of wheels

Demonstrated the conjecture holds for spikes

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.