Embedding cycles of given length in oriented graphs

Daniela Kühn, Deryk Osthus, Diana Piguet

Published 2011-10-25, updated 2012-10-03Version 2

Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed cycle of length l. We prove this conjecture asymptotically for the case when l is large enough compared to k and k>6. The case when k<7 was already settled asymptotically by Kelly, Kuehn and Osthus.