Hamilton cycles in hypergraphs below the Dirac threshold

TL;DR

This paper characterizes when 4-uniform hypergraphs with high codegree contain Hamilton 2-cycles, finds the exact Dirac threshold, and provides a polynomial-time algorithm for detection, contrasting the complexity with graphs.

Contribution

It provides a precise characterization and polynomial-time algorithm for Hamilton 2-cycles in 4-uniform hypergraphs near the Dirac threshold, and shows NP-hardness results for tight Hamilton cycles in general hypergraphs.

Findings

Exact Dirac threshold for Hamilton 2-cycles in 4-uniform hypergraphs.

Polynomial-time algorithm for detecting Hamilton 2-cycles.

NP-hardness of detecting tight Hamilton cycles in general hypergraphs.

Abstract

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.