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

TL;DR

This paper proves that 3-graphs with sufficiently large minimum codegree contain the square of a tight Hamilton cycle, advancing previous bounds and enhancing understanding of their connectivity structure.

Contribution

It improves the minimum codegree threshold for guaranteeing the square of a tight Hamilton cycle in 3-graphs, strengthening prior results.

Findings

Proved that $ ext{min codegree} \, ext{≥} \, 7n/9 + o(n)$ ensures the cycle.

Established a new understanding of connectivity in 3-graphs with large codegree.

Strengthened previous threshold from 4n/5 + o(n) to 7n/9 + o(n).

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.