Minimum vertex degree condition for tight Hamiltonian cycles in 3-uniform hypergraphs

TL;DR

This paper proves that a 3-uniform hypergraph with sufficiently high minimum vertex degree always contains a tight Hamiltonian cycle, establishing an asymptotically optimal degree condition.

Contribution

It establishes the exact asymptotic minimum vertex degree threshold for the existence of tight Hamiltonian cycles in 3-uniform hypergraphs.

Findings

Minimum degree threshold at (5/9+o(1)) times binomial coefficient for tight Hamiltonian cycles

Degree condition is proven to be asymptotically optimal based on known constructions

Every 3-uniform hypergraph exceeding this threshold contains a tight Hamiltonian cycle

Abstract

We show that every 3-uniform hypergraph with $n$ vertices and minimum vertex degree at least $(5/9+o(1))\binom{n}2$ contains a tight Hamiltonian cycle. Known lower bound constructions show that this degree condition is asymptotically optimal.