Nevinson, Emily (2024). From graph minor theory to 3-dimensions. University of Birmingham. Ph.D.
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Nevinson2024PhD.pdf
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Abstract
This thesis comprises of three projects from topological combinatorics and structural graph theory. Firstly, we extend Heawood's theorem on the colourability of plane triangulations to triangulations of 3-space by proving that a triangulation of 3-space can be edge coloured with three colours if and only if all edges have even degree. Next, we propose an open question that seeks to generalise the Four Colour Theorem from two to three dimensions and show that 12 instead of four colours are both sufficient and necessary to colour every 2-complex that embeds in a prescribed 3-manifold. However, our example of a 2-complex that requires 12 colours is not simplicial. We give bounds on this colouring number for simplicial 2-complexes. Lastly, we look at graphs that are not 1-tough and consider the set of minimal seperators of these graphs that "witness" the non-toughness of the graph.
| Type of Work: | Thesis (Doctorates > Ph.D.) | |||||||||
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| Award Type: | Doctorates > Ph.D. | |||||||||
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| Licence: | Creative Commons: Attribution-Noncommercial 4.0 | |||||||||
| College/Faculty: | Colleges > College of Engineering & Physical Sciences | |||||||||
| School or Department: | School of Mathematics | |||||||||
| Funders: | Engineering and Physical Sciences Research Council | |||||||||
| Subjects: | Q Science > QA Mathematics | |||||||||
| URI: | http://etheses.bham.ac.uk/id/eprint/15482 |
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