A Dirac type result on Hamilton cycles in oriented graphs

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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. 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. We also prove an Ore-type theorem for oriented graphs.