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arXiv · 2504.09794

Arbitrary orientations of cycles in oriented graphs

Guanghui Wang·Yun Wang·Zhiwei Zhang
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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.

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Guanghui Wang, Yun Wang, Zhiwei Zhang. 2025-04-14. Arbitrary orientations of cycles in oriented graphs. https://arxiv.org/abs/2504.09794

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