Highly connected matroids are consistently useful in the analysis of matroid structure. Round matroids, in particular, were instrumental in the proof of Rota's conjecture. Chapter 2 concerns a class of matroids with similar properties to those of round matroids. We provide many useful characterizations of these matroids, and determine explicitly their regular members.
Tutte proved that a 3-connected matroid with every element in a 3-element circuit and a 3-element cocircuit is either a whirl or the cycle matroid of a wheel. This result led to the proof of the 3-connected splitter theorem. More recently, Miller proved that matroids of sufficient size having every pair of elements in a 4-element circuit and a 4-element cocircuit are spikes. This observation simplifies the proof of Rota's conjecture for GF(4). In Chapters 3 and 4, we investigate matroids having similar restrictions on their small circuits and cocircuits. The main result of each of these chapters is a complete characterization of the matroids therein.
| Identifer | oai:union.ndltd.org:LSU/oai:etd.lsu.edu:etd-06302016-110350 |
| Date | 21 July 2016 |
| Creators | Pfeil, Simon |
| Contributors | Oxley, James, Oporowski, Bogdan, Baldridge, Scott, Ding, Guoli, Delzell, Charles, McCarter, Kevin |
| Publisher | LSU |
| Source Sets | Louisiana State University |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lsu.edu/docs/available/etd-06302016-110350/ |
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