The regularity method for graphs and digraphs

This MSci thesis surveys results in extremal graph theory, in particular relating to Hamilton cycles. Szem\'eredi's Regularity Lemma plays a central role. We also investigate the robust outexpansion property for digraphs. Kelly showed that every sufficiently large oriented graph on $n$ vertices with minimum in- and outdegree at least $3n/8 +o(n)$ contains any orientation of a Hamilton cycle. We use Kelly's arguments to extend his result to any robustly expanding digraph of linear degree.