On the asymptotic proportion of connected matroids

On the asymptotic proportion of connected matroids

Very little is known about the asymptotic behaviour of classes of matroids. We make a number of conjectures about such behaviours. For example, we conjecture that asymptotically almost every matroid: has a trivial automorphism group; is arbitrarily highly connected; and, is not representable over any field. We prove one result: The proportion of labeled n-element matroids that are connected is ...

On the (co)girth of connected matroids ?

This article studies the girth and cogirth problems for connected matroids. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a linear matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose...

On Nonbinary 3-connected Matroids

It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a U2,4minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary 3-connected matroid is in a U2,4-minor. This paper extends Seymour's theorem by pr...

On minor-minimally-connected matroids

On removable series classes in connected matroids

A series class P of a connected matroid M is removable if M \ P is connected. In this paper, we prove that a connected matroid M with r(M) ≥ 2 has at least r(M) + 1 removable series classes. Further, we obtain certain results from which the following result of Oxley and its graph theoretic version follow: If C is a circuit of a connected matroid M with C 6= M such that M \ x is not connected fo...