Cycles of given length in oriented graphs

Authors:
Luke Kelly;Daniela Kühn;Deryk Osthus
Affiliations:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK;School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK;School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
Venue:
Journal of Combinatorial Theory Series B
Year:
2010

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Abstract

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.