Robust expansion and hamiltonicity

Staden, Katherine (2015). Robust expansion and hamiltonicity. University of Birmingham. Ph.D.

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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 \(D\)-regular graph on \(n\) vertices with \(D\) ≥ 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 \(i\)\(^{th}\) largest degree in a sufficiently large graph \(G\) on n vertices is at least a little larger than \(n\)/3 + \(i\) for \(i\) ≤ \(n\)/3, then \(G\) contains the square of a Hamilton cycle.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):