6 research outputs found
On properties of almost all matroids
We give several results about the asymptotic behaviour of matroids. Specifically, almost all matroids are simple and cosimple and, indeed, are 3-connected. This verifies a strengthening of a conjecture of Mayhew, Newman, Welsh, and Whittle. We prove several quantitative results including giving bounds on the rank, a bound on the number of bases, the number of circuits, and the maximum circuit size of almost all matroids. © 2012 Elsevier Inc
On the number of matroids compared to the number of sparse paving matroids
It has been conjectured that sparse paving matroids will eventually
predominate in any asymptotic enumeration of matroids, i.e. that
, where denotes the number of
matroids on elements, and the number of sparse paving matroids. In
this paper, we show that We prove this by arguing that each matroid on elements has a
faithful description consisting of a stable set of a Johnson graph together
with a (by comparison) vanishing amount of other information, and using that
stable sets in these Johnson graphs correspond one-to-one to sparse paving
matroids on elements.
As a consequence of our result, we find that for some ,
asymptotically almost all matroids on elements have rank in the range .Comment: 12 pages, 2 figure
On properties of almost all matroids
AbstractWe give several results about the asymptotic behaviour of matroids. Specifically, almost all matroids are simple and cosimple and, indeed, are 3-connected. This verifies a strengthening of a conjecture of Mayhew, Newman, Welsh, and Whittle. We prove several quantitative results including giving bounds on the rank, a bound on the number of bases, the number of circuits, and the maximum circuit size of almost all matroids