Towards Pósa's Conjecture for $3$-graphs

Abstract

We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree $\hat{I}{´}_2(H)\ge 7n/9+o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $\hat{I}{´}_2(H)\ge 4n/5+o(n)$ is sufficient. The central novelty of our arguments is an improved understanding of the connectivity structure of $3$-graphs with large minimum codegree.

28 pages