Counting Hamiltonian Cycles in Dirac Hypergraphs

TL;DR

This paper proves that dense hypergraphs with high minimum co-degree contain nearly as many Hamiltonian cycles as random hypergraphs, confirming a conjecture and improving previous results in hypergraph cycle enumeration.

Contribution

It establishes a lower bound on the number of Hamiltonian ycles in dense hypergraphs, extending known results and confirming a key conjecture for all relevant parameters.

Findings

Hypergraphs with minimum co-degree > 1/2 have many Hamiltonian ycles.

The number of cycles matches that of a random hypergraph with the same edge probability.

The result improves previous bounds and confirms a conjecture for all ycle parameters.

Abstract

For $0\leq \ell <k$, 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, K\"uhn, and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values $0\leq \ell<k-1$.