Loose Hamiltonicity

Abstract

We study the appearance of Hamilton $\ell$-cycles in dense $k$-uniform hypergraphs when $\ell \le k-2$ 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 $1\le \ell \le k-2$ 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.