Cyclic Orderings of Paving Matroids

TL;DR

This paper proves a conjecture characterizing cyclic orderability of paving matroids, showing a specific ratio condition is both necessary and sufficient for such orderings.

Contribution

It verifies a longstanding conjecture for paving matroids, establishing a clear ratio-based criterion for cyclic orderability.

Findings

The conjecture holds true for all paving matroids.

A ratio condition characterizes cyclic orderability in this class.

The result advances understanding of matroid cyclic orderings.

Abstract

A matroid M is cyclically orderable if there is a cyclic permutation of the elements of M such that any r consecutive elements form a basis in M. An old conjecture of Kajitani, Miyano, and Ueno states that a matroid M is cyclically orderable if and only if for all nonempty subsets X in E(M), |X|/r(M) is less than or equal to |E(M)|/r(M). In this paper, we verify this conjecture for all paving matroids.