# Loose Hamiltonian cycles forced by large $(k-2)$-degree - sharp version

## TL;DR

This paper establishes the exact minimum degree condition needed for large uniform hypergraphs to contain Hamiltonian cycles, extending previous results to higher uniformities.

## Contribution

It provides the sharp $(k-2)$-degree threshold for Hamiltonian $ ext{ell}$-cycles in $k$-uniform hypergraphs, generalizing earlier work for 3-uniform cases.

## Key findings

- Determined the exact degree threshold for Hamiltonian cycles in hypergraphs.
- Extended previous results from 3-uniform to general $k$-uniform hypergraphs.
- Proved the sharpness of the degree condition for sufficiently large hypergraphs.

## Abstract

We prove for all $k\geq 4$ and $1\leq\ell<k/2$ the sharp minimum $(k-2)$-degree bound for a $k$-uniform hypergraph $\mathcal H$ on $n$ vertices to contain a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$ and $n$ is sufficiently large. This extends a result of Han and Zhao for $3$-uniform hypegraphs.