Abstract. We give, for each [Formula: see text], the precise best possible minimum positive codegree condition for a perfect matching in a large [Formula: see text]-uniform hypergraph [Formula: see text] on [Formula: see text] vertices. Specifically, we show that if [Formula: see text] is sufficiently large and divisible by [Formula: see text] and [Formula: see text] has minimum positive codegree [Formula: see text] and no isolated vertices, then [Formula: see text] contains a perfect matching. For [Formula: see text], this was previously established by Halfpap and Magnan [ Positive Co-Degree Thresholds for Spanning Structures, 2024], who also gave bounds for [Formula: see text] which were tight up to an additive constant.

In this paper we prove an asymptotic multipartite version of a well-known theorem of Kühn and Osthus by establishing, for any graph $H$ with chromatic number $r$, the asymptotic multipartite minimum degree threshold which ensures that a large $r$-partite graph $G$ admits a perfect $H$-tiling. We also give the threshold for an $H$-tiling covering all but a linear number of vertices of $G$, in a multipartite analogue of results of Komlós and of Shokoufandeh and Zhao.

We consider the complexity of the Hamilton cycle decision problem when restricted to k-uniform hypergraphs H of high minimum codegree delta(H). We show that for tight Hamilton cycles this problem is NP-hard even when restricted to k-uniform hypergraphs H with delta(H) >= n/2 - C, where n is the order of H and C is a constant which depends only on k. This answers a question raised by Karpinski, Rucinski and Szymanska. Additionally we give a polynomial-time algorithm which, for a sufficiently small constant epsilon > 0, determines whether or not a 4-uniform hypergraph H on n vertices with delta(H) >= n/2 - epsilon * n contains a Hamilton 2-cycle. This demonstrates that some looser Hamilton cycles exhibit interestingly different behaviour compared to tight Hamilton cycles. A key part of the proof is a precise characterisation of all 4-uniform hypergraphs H on n vertices with delta(H) >= n/2 - epsilon * n which do not contain a Hamilton 2-cycle; this may be of independent interest. As an additional corollary of this characterisation, we obtain an exact Dirac-type bound for the existence of a Hamilton 2-cycle in a large 4-uniform hypergraph.

Let H be a k-graph on n vertices, with minimum codegree at least n/k + cn for some fixed c > 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H or a certificate that none exists. This essentially solves a problem of Karpinski, Rucinski and Szymanska, who previously showed that this problem is NP-hard for a minimum codegree of n/k - cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.