Cyclic matroids

TL;DR

This paper characterizes nearly cyclic matroids with large ground sets, showing they are actually cyclic and can be derived from specific well-understood matroids through truncation and weak-map operations.

Contribution

It establishes the structure of nearly $(s,t)$-cyclic matroids for large $n$, proving they are $(s,t)$-cyclic and related to truncated whirl and swirl matroids.

Findings

Nearly $(s,t)$-cyclic implies $(s,t)$-cyclic for large $n$

$(s,t)$-cyclic matroids are weak-map images of truncated well-known matroids

Special cases include whirl and swirl matroids for $s=3$ and $s=4$

Abstract

For all positive integers $s$ and $t$ exceeding one, a matroid $M$ on $n$ elements is {\em nearly $(s, t)$-cyclic} if there is a cyclic ordering $\sigma$ of its ground set such that every $s-1$ consecutive elements of $\sigma$ are contained in an $s$-element circuit and every $t-1$ consecutive elements of $\sigma$ are contained in a $t$-element cocircuit. In the case $s=t$, nearly $(s, s)$-cyclic matroids have been studied previously. In this paper, we show that if $M$ is nearly $(s, t)$-cyclic and $n$ is sufficiently large, then these $s$-element circuits and $t$-element cocircuits are consecutive in $\sigma$ in a prescribed way, that is, $M$ is "$(s, t)$-cyclic". Furthermore, we show that, given $s$ and $t$ where $t\ge s$, every $(s, t)$-cyclic matroid on $n > s+t-2$ elements is a weak-map image of the $\left(\frac{t-s}{2}\right)$-th truncation of a certain $(s, s)$-cyclic matroid. If $s=3$, this certain matroid is the rank-$\frac{n}{2}$ whirl, and if $s=4$, this certain matroid is the rank-$\frac{n}{2}$ free swirl.