Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree
Combinatorics
2025-05-16 v1
Abstract
In 1960, Ghouila-Houri proved that every strongly connected directed graph on vertices with minimum degree at least contains a directed Hamilton cycle. We asymptotically generalize this result by proving the following: every directed graph on vertices and with minimum degree at least contains every orientation of a Hamilton cycle, except for the directed Hamilton cycle in the case when is not strongly connected. In fact, this minimum degree condition forces every orientation of a cycle in of every possible length, other than perhaps the directed cycles.
Cite
@article{arxiv.2505.09793,
title = {Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree},
author = {Louis DeBiasio and Andrew Treglown},
journal= {arXiv preprint arXiv:2505.09793},
year = {2025}
}
Comments
18 pages (plus 5 page appendix), 2 figures