Loose Hamiltonicity

TL;DR

This paper investigates Hamilton $ ext{ell}$-cycles in dense $k$-uniform hypergraphs, providing thresholds and reductions for their existence, and introduces new methods avoiding traditional complex techniques.

Contribution

It reduces the problem of Hamilton $ ext{ell}$-cycles to cycle tiling problems in host graphs and determines degree thresholds for various parameters, using novel blow-up cover methods.

Findings

Established degree thresholds for Hamilton $ ext{ell}$-cycles in hypergraphs.

Reduced complex cycle existence problems to tiling problems in host graphs.

Introduced blow-up cover method avoiding Regularity Lemma and Absorption Method.

Abstract

We study the appearance of Hamilton $\ell$-cycles in dense $k$-uniform hypergraphs when $\ell \leq k-2$ and $k-\ell$ does not divide $k$. Our main result reduces this problem to the robust existence of a connected $\ell$-cycle tiling in host graph families that are approximately closed under subsampling. As an application, we determine the minimum $d$-degree threshold for $d=k-2$ and all $1 \leq \ell \leq k-2$ when $k - \ell$ does not divide $k$. We also reduce the case $\ell < d$ entirely to the corresponding (non-connected) $\ell$-cycle tiling problem. In addition, our outcomes lead to counting and random robust versions of these results. The proofs are based on the recently introduced method of blow-up covers and thus avoid the use of the Regularity Lemma and the Absorption Method.