On circuits and serial symmetric basis-exchange in matroids

TL;DR

This paper investigates how circuits change during basis exchanges in matroids, proving the existence of symmetric exchanges and providing a new characterization of binary matroids.

Contribution

It establishes the existence of three consecutive symmetric exchanges between any two bases and introduces a new basis-exchange characterization of binary matroids.

Findings

Three consecutive symmetric exchanges always exist.

A full serial symmetric exchange of length at most 6 exists for matroids of rank 5.

A new characterization of binary matroids related to basis exchange.

Abstract

The way circuits, relative to a basis, are affected as a result of exchanging a basis element, is studied. As consequences, it is shown that three consecutive symmetric exchanges exist for any two bases of a matroid, and that a full serial symmetric exchange, of length at most 6, exists for any two bases of a matroid of rank 5. A new characterization of binary matroids, related to basis-exchange, is presented.