Embedding cycles of given length in oriented graphs

TL;DR

This paper proves an asymptotic version of a conjecture on embedding cycles of specified length in large oriented graphs with high minimum degree, extending previous results for smaller values of k.

Contribution

It establishes the conjecture asymptotically for large cycle lengths when k>6, filling a gap in the understanding of cycle embeddings in oriented graphs.

Findings

Proves the Kelly-Kuehn-Osthus conjecture asymptotically for large l and k>6.

Extends previous results to larger cycle lengths and k values.

Provides new techniques for embedding cycles in dense oriented graphs.

Abstract

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.