Counting spanning subgraphs in dense hypergraphs

TL;DR

This paper introduces a straightforward method to estimate the number of certain spanning subgraphs, like Hamilton cycles, in dense hypergraphs with high minimum degree, providing new bounds and simpler proofs for existing results.

Contribution

It presents a simple estimation technique for counting spanning subgraphs in dense hypergraphs, improving bounds and simplifying proofs for Hamilton cycle counts under high minimum codegree conditions.

Findings

Estimates the number of Hamilton $ ext{ell}$-cycles in dense hypergraphs.

Provides asymptotically optimal bounds for minimum codegree conditions.

Simplifies proofs of existing results on spanning subgraphs in hypergraphs.

Abstract

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell\leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least $$\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\ \left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if $(k-\ell)\nmid k$,}$$ contains $\exp(n\log n-\Theta(n))$ Hamilton $\ell$-cycles as long as $(k-\ell)\mid n$. When $(k-\ell)\mid k$ this gives a simple proof of a result of Glock, Gould, Joos, K\"uhn and Osthus, while, when $(k-\ell)\nmid k$ this gives a weaker count than that given by Ferber, Hardiman and Mond or, when $\ell<k/2$, by Ferber, Krivelevich and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.