The Asymptotic Number of Connected d-Uniform Hypergraphs

Abstract

For d ≥ 2, let Hd(n,p) denote a random d-uniform hypergraph with n vertices in which each of the $\binom{n}{d}$ possible edges is present with probability p=p(n) independently, and let Hd(n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. Let either H=Hd(n,m) or H=Hd(n,p), where m/n and $\binom{n-1}{d-1}p$ need to be bounded away from (d−1)−1 and 0 respectively. We determine the asymptotic probability that H is connected. This yields the asymptotic number of connected d-uniform hypergraphs with given numbers of vertices and edges. We also derive a local limit theorem for the number of edges in Hd(n,p), conditioned on Hd(n,p) being connected.