Cycles of given length in oriented graphs

We show that for each @?>=4 every sufficiently large oriented graph G with @d^+(G),@d^-(G)>=@?|G|/3@?+1 contains an @?-cycle. This 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.

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