On Hamiltonian cycles in hypergraphs with dense link graphs

TL;DR

This paper proves that dense enough hypergraphs with a specific minimum degree condition always contain a Hamiltonian cycle, confirming the optimality of this threshold.

Contribution

It establishes a tight minimum degree condition for the existence of Hamiltonian cycles in k-uniform hypergraphs, matching previous constructions.

Findings

Minimum degree threshold of (5/9+o(1))n^2/2 guarantees Hamiltonian cycles

The result confirms the optimality of the degree condition

Independent proof by Lang and Sahueza-Matamala

Abstract

We show that every $k$-uniform hypergraph on $n$ vertices whose minimum $(k-2)$-degree is at least $(5/9+o(1))n^2/2$ contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sahueza-Matamala.