Towards P\'osa's Conjecture for $3$-graphs

Debmalya Bandyopadhyay; Allan Lo; Richard Mycroft

Towards P\'osa's Conjecture for $3$-graphs

Combinatorics 2026-03-31 v1

Authors: Debmalya Bandyopadhyay , Allan Lo , Richard Mycroft

Abstract

We prove that every $3$ -graph $H$ on $n$ vertices with minimum codegree $\delta_2(H) \geq 7n/9 + o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $\delta_2(H) \geq 4n/5 + o(n)$ is sufficient. The central novelty of our arguments is an improved understanding of the connectivity structure of $3$ -graphs with large minimum codegree.

Keywords

hamilton cycles graph cycles hadwiger conjecture

Cite

@article{arxiv.2603.28202,
  title  = {Towards P\'osa's Conjecture for $3$-graphs},
  author = {Debmalya Bandyopadhyay and Allan Lo and Richard Mycroft},
  journal= {arXiv preprint arXiv:2603.28202},
  year   = {2026}
}

Comments

28 pages

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