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arXiv · 2505.09793

Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree

Louis DeBiasio·Andrew Treglown
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Abstract

In 1960, Ghouila-Houri proved that every strongly connected directed graph $G$ on $n$ vertices with minimum degree at least $n$ contains a directed Hamilton cycle. We asymptotically generalize this result by proving the following: every directed graph $G$ on $n$ vertices and with minimum degree at least $(1+o(1))n$ contains every orientation of a Hamilton cycle, except for the directed Hamilton cycle in the case when $G$ is not strongly connected. In fact, this minimum degree condition forces every orientation of a cycle in $G$ of every possible length, other than perhaps the directed cycles.

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Louis DeBiasio, Andrew Treglown. 2025-05-14. Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree. https://arxiv.org/abs/2505.09793

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