Two generalisations of the wheels-and-whirls theorem.
| dc.contributor.author | Toft, Gerry | |
| dc.date.accessioned | 2023-12-10T23:40:07Z | |
| dc.date.available | 2023-12-10T23:40:07Z | |
| dc.date.issued | 2023 | |
| dc.description.abstract | One of the most famous results in matroid theory is Tutte’s Wheels-and-Whirls Theorem. It states that every 3-connected matroid has an element which can either be deleted or con- tracted while retaining 3-connectivity, except for two families of matroids: the eponymous wheels and whirls. The Wheels-and-Whirls Theorem is a powerful tool for inductive argu- ments on 3-connected matroids. We consider two generalisations of the Wheels-and-Whirls Theorem. First, what are the k-connected matroids such that the deletion and contraction of every element is not k-connected? Motivated by this problem, we consider matroids in which every element is contained in a small circuit and a small cocircuit, and, in particular, when these circuits and cocircuits have a cyclic structure. The first part of this thesis is concerned with matroids in which have a cyclic ordering σ of their ground set such that every set of s − 1 consecutive elements of σ is contained in an s-element circuit and every set of t − 1 consecutive elements of σ is contained in a t-element circuit. We show that these matroids are highly structured by proving that they are “(s, t)-cyclic”, that is, their s-element circuits and t-element cocircuits are consecutive in σ in a prescribed way. Next, we provide a characterisation of these matroids by showing that every (s, t)-cyclic matroid is a weak-map image of a particular (s, t)-cyclic matroid. Secondly, what are the 3-connected matroids such that such that the deletion and con- traction of every 2-element subset is not 3-connected? In the second part of this thesis, we find all such matroids. Roughly speaking, these matroids can be constructed in one of four ways: by attaching fans to a spike, by attaching fans to a line, by attaching particular matroids to M (K3,m), or by attaching particular matroids to each end of a fan. | |
| dc.identifier.uri | https://hdl.handle.net/10092/106519 | |
| dc.identifier.uri | https://doi.org/10.26021/15157 | |
| dc.language | English | |
| dc.language.iso | en | |
| dc.rights | All Right Reserved | |
| dc.rights.uri | https://canterbury.libguides.com/rights/theses | |
| dc.title | Two generalisations of the wheels-and-whirls theorem. | |
| dc.type | Theses / Dissertations | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | University of Canterbury | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy | |
| uc.college | Faculty of Engineering |
Files
License bundle
1 - 1 of 1
No Thumbnail Available
- Name:
- license.txt
- Size:
- 1.71 KB
- Format:
- Item-specific license agreed upon to submission
- Description: