English

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

Keywords

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