On Serial Symmetric Exchanges of Matroid Bases

TL;DR

This paper investigates serial symmetric exchanges between bases in matroids, proving that any two elements of one base can be exchanged with two elements of another, especially in rank 4 matroids, revealing new exchange properties.

Contribution

It introduces new properties of serial symmetric exchanges in matroids and proves that any two bases of rank 4 can be fully exchanged in a serial manner.

Findings

Any two elements of one base can be serially symmetrically exchanged with two elements of another base.

Any two disjoint bases in a rank 4 matroid admit a full serial symmetric exchange.

Abstract

We study some properties of a serial (i.e. one-by-one) symmetric exchange of elements of two disjoint bases of a matroid. We show that any two elements of one base have a serial symmetric exchange with some two elements of the other base. As a result, we obtain that any two disjoint bases in a matroid of rank 4 have a full serial symmetric exchange.