UWSpace is currently experiencing technical difficulties resulting from its recent migration to a new version of its software. These technical issues are not affecting the submission and browse features of the site. UWaterloo community members may continue submitting items to UWSpace. We apologize for the inconvenience, and are actively working to resolve these technical issues.
 

Modularity and Structure in Matroids

Loading...
Thumbnail Image

Date

2013-04-22T22:39:13Z

Authors

Kapadia, Rohan

Journal Title

Journal ISSN

Volume Title

Publisher

University of Waterloo

Abstract

This thesis concerns sufficient conditions for a matroid to admit one of two types of structural characterization: a representation over a finite field or a description as a frame matroid. We call a restriction N of a matroid M modular if, for every flat F of M, r_M(F) + r(N) = r_M(F ∩ E(N)) + r_M(F ∪ E(N)). A consequence of a theorem of Seymour is that any 3-connected matroid with a modular U_{2,3}-restriction is binary. We extend this fact to arbitrary finite fields, showing that if N is a modular rank-3 restriction of a vertically 4-connected matroid M, then any representation of N over a finite field extends to a representation of M. We also look at a more general notion of modularity that applies to minors of a matroid, and use it to present conditions for a matroid with a large projective geometry minor to be representable over a finite field. In particular, we show that a 3-connected, representable matroid with a sufficiently large projective geometry over a finite field GF(q) as a minor is either representable over GF(q) or has a U_{2,q^2+1}-minor. A second result of Seymour is that any vertically 4-connected matroid with a modular M(K_4)-restriction is graphic. Geelen, Gerards, and Whittle partially generalized this from M(K_4) to larger frame matroids, showing that any vertically 5-connected, representable matroid with a rank-4 Dowling geometry as a modular restriction is a frame matroid. As with projective geometries, we prove a version of this result for matroids with large Dowling geometries as minors, providing conditions which imply that they are frame matroids.

Description

Keywords

matroids, combinatorics

LC Keywords

Citation