Abstract. We show that for each α > 0 every sufficiently large oriented graph G with δ + (G), δ − (G) ≥ 3|G|/8 + α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen [21]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G) + δ + (G) + δ − (G) ≥ 3|G|/2 + α|G|. Up to the term α|G| this confirms a conjecture of Häggkvist [10]. We also prove an Ore-type theorem for oriented graphs.

We show that for each 4 every sufficiently large orientedThis is best possible for all those 4 which are not divisible by 3. Surprisingly, for some other values of , an -cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an -cycle (with 4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity and consider -cycles in general digraphs.

We use a randomised embedding method to prove that for all α > 0 any sufficiently large oriented graph G with minimum in-degree and out-degree δ + (G), δ − (G) ≥ (3/8 + α)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of Häggkvist and Thomason.

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