Arbitrary orientations of cycles in oriented graphs

Abstract

We show that every sufficiently large oriented graph $G$ with minimum indegree and outdegree both at least $(3|V(G)|-1)/8$ contains every orientation of a Hamilton cycle. This result improves the approximate bound established by Kelly and resolves a long-standing problem posed by H\"aggkvist and Thomason in 1995. The degree condition is tight and it can be improved to $(3|V(G)|-4)/8$ for Hamilton cycles that are nearly directed, generalizing a classic result by Keevash, K\"uhn and Osthus. Additionally, we derive a pancyclicity result for arbitrary orientations. More precisely, the above degree condition suffices to guarantee the existence of cycles of every possible orientation and every possible length unless $G$ is isomorphic to one of the exceptional oriented graphs.