Counting Hamilton cycles in Dirac hypergraphs

Abstract

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. Rödl, Ruciński, and Szemerédi proved that for $k\ge 3$, every $k$-graph on $n$ vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such $k$-graphs is $\exp (n\ln n-\Theta (n))$. As a corollary, we obtain a similar estimate on the number of Hamilton $\ell$-cycles in such $k$-graphs for all $\ell \in \{0,\dots ,k-1\}$, which makes progress on a question of Ferber, Krivelevich and Sudakov.

20 pages. Final version, to appear in Combinatorics, Probability & Computing