Maximal Circuits in Matroids.

Lovasz, Schrijver, and Seymour have shown that if a connected matroid M has a largest circuit of size c and a largest cocircuit of size c*, then M has at most 2c+c*-1 elements. A question of Oxley that has attracted considerable recent attention is whether there is an upper bound on the size of M that is polynomial in terms of c and c*. In particular, Bonin, McNulty, and Reid conjectured that 12cc* is such a bound. In Chapter 1, we prove this conjecture for an connected graphic and cographic matroids. In Chapter 2, we give a constructive description of all 2-connected graphs that attain equality in this bound showing that these graphs are certain special series-parallel networks. In Chapter 3, we investigate whether in a k-connected matroid M with a large circuit there is a large circuit containing n specified elements. Assume that the size of a largest circuit in M is c for some c ≥ 4. We prove that, for k ∈ {2, 3}, every element of M is contained in a circuit of size at least &ceill0;c2&ceilr0;+k-1. Even when M is 3-connected and binary, the presence of a large circuit in M does not guarantee that M has a large circuit containing a nominated pair of elements. However, when M is 3-connected and graphic, it will be shown that every pair of distinct elements is contained in a circuit of size at least &ceill0;c-2&ceilr0;+2. Examples will be given to show that these results are best-possible. A result of Ding, Oporowski, Oxley, and Vertigan shows that if C is a largest circuit of a 3-connected matroid M, then M has a 3-connected minor N in which C is a spanning circuit of N. We extend this result by showing that the 3-connected minor N that is spanned by C can also be required to contain a specified element. This extension plays a key role in the proofs of the main results of this chapter, which were noted above

Ramsey Theory Using Matroid Minors

This thesis considers a Ramsey Theory question for graphs and regular matroids. Specifically, how many elements N are required in a 3-connected graphic or regular matroid to force the existence of certain specified minors in that matroid? This question cannot be answered for an arbitrary collection of specified minors. However, there are results from the literature for which the number N exists for certain collections of minors. We first encode totally unimodular matrix representations of certain matroids. We use the computer program MACEK to investigate this question for certain classes of specified minors