Hamilton cycles in hypergraphs below the Dirac threshold

We establish a precise characterisation of $4$-uniform hypergraphs with minimum codegree close to $n/2$ which contain a Hamilton $2$-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton $2$-cycles in $4$-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a $4$-uniform hypergraph $H$ with minimum codegree close to $n/2$, either finds a Hamilton $2$-cycle in $H$ or provides a certificate that no such cycle exists. This surprising result stands in contrast to the graph setting, in which below the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We also consider tight Hamilton cycles in $k$-uniform hypergraphs $H$ for $k \geq 3$, giving a series of reductions to show that it is NP-hard to determine whether a $k$-uniform hypergraph $H$ with minimum degree $\delta(H) \geq \frac{1}{2}|V(H)| - O(1)$ contains a tight Hamilton cycle. It is therefore unlikely that a similar characterisation can be obtained for tight Hamilton cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode and details of the polynomial time reduction moved to the appendix which will not appear in the printed version of the paper. To appear in Journal of Combinatorial Theory, Series