Asymptotic enumeration of sparse uniform hypergraphs with given degrees

TL;DR

This paper derives an asymptotic formula for counting simple r-uniform hypergraphs with a specified degree sequence, applicable under certain maximum degree constraints, extending enumeration methods in hypergraph theory.

Contribution

It provides a new asymptotic enumeration formula for sparse uniform hypergraphs with given degrees, under specific maximum degree conditions.

Findings

Derived an asymptotic enumeration formula for hypergraphs

Applicable to hypergraphs with maximum degree satisfying $k_{max}^3 = o(M)$

Extends enumeration techniques to sparse hypergraph classes

Abstract

Let $r \geq 2$ be a fixed integer. For infinitely many $n$, let $\boldsymbol{k} = (k_1,..., k_n)$ be a vector of nonnegative integers such that their sum $M$ is divisible by $r$. We present an asymptotic enumeration formula for simple $r$-uniform hypergraphs with degree sequence $k$. (Here "simple" means that all edges are distinct and no edge contains a repeated vertex.) Our formula holds whenever the maximum degree $k_{\mathrm{max}}$ satisfies $k_{\mathrm{max}}^3 = o(M)$.