A Dirac-Type Result on Hamilton Cycles in Oriented Graphs

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.