Infinite graphic matroids Part I

An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graph-like topological space: that is, a graph-like space in the sense of Thomassen and Vella. This extends Tutte's characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3-connected graphic matroid is countable

Infinite trees of matroids

We generalise the construction of infinite matroids from trees of matroids to allow the matroids at the nodes, as well as the field over which they are represented, to be infinite

On the intersection conjecture for infinite trees of matroids

Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts

Axioms for infinite matroids

We give axiomatic foundations for non-finitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig

Graph Theory

This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion

On the Matroid Intersection Conjecture

In this dissertation, we investigate the Matroid Intersection Conjecture for pairs of matroids on the same ground set, proposed by Nash-Williams in 1990. Originally, the conjecture was stated for finitary matroids only, but we consider it for general matroids and introduce new approaches to attack the conjecture.;The first approach is to consider the situation when it is possible to make a finite modification to the matroids after which the pair satisfies the conjecture. In such a situation we say that the pair has the Almost Intersection Property. We prove that any pair of matroids with the Almost Intersection Property must satisfy the Matroid Intersection Conjecture. Using this result we prove that the Matroid Intersection Conjecture is true in the case when one of the matroids has finite rank and also in the case when one of the matroids is a patchwork matroid.;Our second new approach is inspired by the proof of the general version of Konig\u27s Theorem for bipartite graphs. That result implies that the Matroid Intersection Conjecture is true for pairs of partition matroids. We develop some new techniques that generalize the critical set approach used in the proof of the countable version of Konig\u27s Theorem. Our results enable us to prove that the Matroid Intersection Conjecture is true for a pair of singular matroids on a set that is infinitely countable. A matroid is singular when it is a direct sum of matroids such that each term of the sum is a uniform matroid either of rank one or of co-rank one