Staden, Katherine (2015). Robust expansion and hamiltonicity. University of Birmingham. Ph.D.
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Staden15PhD.pdf
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Abstract
This thesis contains four results in extremal graph theory relating to the recent notion of robust expansion, and the classical notion of Hamiltonicity. In Chapter 2 we prove that every sufficiently large ‘robustly expanding’ digraph which is dense and regular has an approximate Hamilton decomposition. This provides a common generalisation of several previous results and in turn was a crucial tool in Kühn and Osthus’s proof that in fact these conditions guarantee a Hamilton decomposition, thereby proving a conjecture of Kelly from 1968 on regular tournaments.
In Chapters 3 and 4, we prove that every sufficiently large 3-connected -regular graph on vertices with ≥ n/4 contains a Hamilton cycle. This answers a problem of Bollobás and Häggkvist from the 1970s. Along the way, we prove a general result about the structure of dense regular graphs, and consider other applications of this.
Chapter 5 is devoted to a degree sequence analogue of the famous Pósa conjecture. Our main result is the following: if the largest degree in a sufficiently large graph on n vertices is at least a little larger than /3 + for ≤ /3, then contains the square of a Hamilton cycle.
Type of Work: | Thesis (Doctorates > Ph.D.) | |||||||||
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Award Type: | Doctorates > Ph.D. | |||||||||
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College/Faculty: | Colleges (2008 onwards) > College of Engineering & Physical Sciences | |||||||||
School or Department: | School of Mathematics | |||||||||
Funders: | None/not applicable | |||||||||
Subjects: | Q Science > QA Mathematics | |||||||||
URI: | http://etheses.bham.ac.uk/id/eprint/5981 |
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