Counting Hamiltonian Cycles in Dirac Hypergraphs

Abstract

For $0\le \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We show that for all $0\le \ell <k-1$, every $k$-graph with minimum co-degree $\delta n$ with $\delta >1/2$ has (asymptotically and up to a subexponential factor) at least as many Hamiltonian $\ell$-cycles as in a typical random $k$-graph with edge-probability $\delta$. This significantly improves a recent result of Glock, Gould, Joos, Kühn, and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values $0\le \ell <k-1$.

14 pages