Matroid multiple cyclic exchange property

TL;DR

This paper introduces a new exchange property for matroid bases that generalizes existing symmetric exchange properties, involving cyclic shifts of subsets to produce bases.

Contribution

It proves a novel cyclic exchange property for matroid bases that extends the multiple symmetric exchange property.

Findings

Established a new cyclic exchange property for matroid bases

Generalized the multiple symmetric exchange property

Provided a constructive proof for the exchange property

Abstract

We prove a new exchange property for bases of a matroid that generalizes the multiple symmetric exchange property. For every bases $B_1,\dots,B_k$ of a matroid and a subset $A_1\subset B_1$ there exist subsets $A_2\subset B_2,\dots,A_k\subset B_k$ such that all sets $(B_i\setminus A_i)\cup A_{i-1}$ achieved by a cyclic shift of $A_i$'s by one are bases.