Cyclic matroids

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.

29 pages, 1 figure