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

TL;DR

This paper establishes near-optimal degree conditions in large hypergraphs that guarantee the existence of Hamiltonian cycles, extending previous results to a broader range of hypergraph parameters.

Contribution

It proves an asymptotically best possible degree threshold for Hamiltonian -cycles in hypergraphs with large minimum -degree, generalizing prior work.

Findings

Degree condition is asymptotically optimal.

Hypergraphs with specified minimum -degree contain Hamiltonian cycles.

Extends known results to larger uniformities.

Abstract

We prove that for all $k\geq 4$ and $1\leq\ell<k/2$, every $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with $\delta_{k-2}(\mathcal{H})\geq\left(\frac{4(k-\ell)-1}{4(k-\ell)^2}+o(1)\right)\binom{n}{2}$ contains a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$. This degree condition is asymptotically best possible. The case $k=3$ was addressed earlier by Bu{\ss} et al.