Cyclic Matroids

Abstract

For integers and exceeding one, a matroid on elements is nearly -cyclic if there is a cyclic ordering of its ground set such that every consecutive elements of are contained in an -element circuit and every consecutive elements of are contained in a -element cocircuit. In the case , nearly -cyclic matroids have been studied previously. In this paper, we show that if is nearly -cyclic and is sufficiently large, then these -element circuits and -element cocircuits are consecutive in in a prescribed way, that is, is “ -cyclic.” Furthermore, we show that, given and where , every -cyclic matroid on elements is a weak-map image of the th truncation of a certain -cyclic matroid. If , this certain matroid is the rank- whirl, and if , this certain matroid is the rank- free swirl.