Date of Award

1998

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

James Oxley

Abstract

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.

ISBN

9780599213333

Pages

76

DOI

10.31390/gradschool_disstheses.6879

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