Cyclic ordering of split matroids

TL;DR

This paper proves that split matroids with disjoint bases are cyclically orderable, providing an algorithmic method to find such orderings and addressing related conjectures in matroid theory.

Contribution

It establishes that split matroids with disjoint bases are cyclically orderable and offers a polynomial-time algorithm to find the cyclic ordering.

Findings

Split matroids with disjoint bases are cyclically orderable.

Provides a polynomial-time algorithm for constructing cyclic orderings.

Answers a special case of a conjecture by Kajitani, Ueno, and Miyano.

Abstract

There is a long list of open questions rooted in the same underlying problem: understanding the structure of bases or common bases of matroids. These conjectures suggest that matroids may possess much stronger structural properties than are currently known. One example is related to cyclic orderings of matroids. A rank-$r$ matroid is called cyclically orderable if its ground set admits a cyclic ordering such that any interval of $r$ consecutive elements forms a basis. In this paper, we show that if the ground set of a split matroid decomposes into pairwise disjoint bases, then it is cyclically orderable. This result answers a conjecture of Kajitani, Ueno, and Miyano in a special case, and also strengthens Gabow's conjecture for this class of matroids. Our proof is algorithmic, hence it provides a procedure for determining a cyclic ordering in question using a polynomial number of independence oracle calls.