Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs

TL;DR

This paper establishes the exact minimum vertex degree threshold for the existence of loose Hamilton cycles in large 3-uniform hypergraphs, improving previous asymptotic results with precise bounds.

Contribution

It provides the exact degree condition for large hypergraphs to contain loose Hamilton cycles, refining prior asymptotic findings with sharp thresholds.

Findings

Degree threshold is optimal and exact for large hypergraphs.

The result improves upon previous asymptotic bounds.

The threshold depends on the divisibility properties of n.

Abstract

We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$, where $c=2$ if $n\in 4\mathbb{N}$ and $c=1$ if $n\in 2\mathbb{N}\setminus 4\mathbb{N}$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Bu\ss, H\`an and Schacht who proved the corresponding asymptotical result.