Matroids with a cyclic arrangement of circuits and cocircuits

TL;DR

This paper characterizes the structure of matroids with a cyclic arrangement of circuits and cocircuits, revealing patterns similar to wheels, whirls, and swirls, and introduces the concept of flowers in these arrangements.

Contribution

It establishes the prescribed arrangement of circuits and cocircuits in matroids with the cyclic property and classifies the resulting flowers as daisies or anemones depending on parity.

Findings

For large matroids with the cyclic property, circuits and cocircuits follow a specific arrangement.

The arrangement resembles known structures like wheels, whirls, and swirls.

Concatenations of the cyclic ordering form flowers, which are daisies or anemones depending on t.

Abstract

For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $\sigma$ such that every set of $t-1$ consecutive elements in $\sigma$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $\sigma$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element circuits and cocircuits appear in swirls. Furthermore, we show that any appropriate concatenation $\Phi$ of $\sigma$ is a flower. If $t$ is odd, then $\Phi$ is a daisy, but if $t$ is even, then, depending on $M$, it is possible for $\Phi$ to be either an anemone or a daisy.