The Complexity of the Hamilton Cycle Problem in Hypergraphs of High Minimum Codegree

We consider the complexity of the Hamilton cycle decision problem when restricted to k-uniform hypergraphs H of high minimum codegree δ(H). We show that for tight Hamilton cycles this problem is NP-hard even when restricted to k-uniform hypergraphsH with δ(H) ≥ n2−C, where n is the order of H and C is a constant which depends only on k. This answers a question raised by Karpiński, Ruciński and Szymańska. Additionally we give a polynomial-time algorithm which, for a sufficiently small constant ε > 0, determines whether or not a 4-uniform hypergraph H on n vertices with δ(H) ≥ n2 − εn 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 δ(H) ≥ n2 −εn 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. 1998 ACM Subject Classification G.2.2 Graph Theory – Hypergraphs, Graph Algorithms