Introduction

Dirac's theorem states that every graph G with minimum degree δ(G) ≥ |G|/2 has a hamilton cycle. The simplest analogue for digraphs is given by the theorem of Ghouila-Houri. Given a digraph G of order n and a vertex v ∈ G, we denote the outdegree of v by d+(v) and the indegree by d−(v). We also define d°(v) to be min{d+(v), d−(v)}, and δ°(G) to be min{d°(v): v ∈ G}. Ghouila-Houri's theorem implies that G contains a directed hamilton cycle if d°(G)(G) ≥ n/2. Only recently has a constant c < ½ been established such that every oriented graph satisfying δ°(G) > cn has a directed hamilton cycle; Häggkvist has shown that c = (½ − 2−15) will suffice. He also showed that the condition δ°(G) > n/3 proposed by Thomassen is inadequate to guarantee a hamilton cycle, and conjectured that δ°(G) ≥ 3n/8 is sufficient.

When considering hamilton cycles in digraphs there is no reason to stick to directed cycles only; we might ask for any orientation of an n-cycle. For tournaments G, Thomason has shown that G will contain every oriented cycle (except the directed cycle if G is not strong) regardless of the degrees, provided n is large.