Counting Hamilton cycles in Dirac hypergraphs

TL;DR

This paper establishes an exponential estimate for the number of tight Hamilton cycles in Dirac hypergraphs, advancing understanding of their combinatorial structure and answering a question about Hamilton $ ext{ell}$-cycles.

Contribution

It provides the first exponential estimate on the count of tight Hamilton cycles in Dirac hypergraphs, extending to Hamilton $ ext{ell}$-cycles for all $ ext{ell}$, thus making significant progress on a known open problem.

Findings

Number of tight Hamilton cycles is $ ext{exp}(n ext{ln} n - ext{Theta}(n))$ in Dirac hypergraphs.

Derived estimates for the number of Hamilton $ ext{ell}$-cycles for all $ ext{ell} ext{ in}\{0, ext{...},k-1 ext{}",

Progress on a question by Ferber, Krivelevich, and Sudakov regarding Hamilton cycles.

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. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved that for $k\geq 3$, 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.