Counting dense connected hypergraphs via the probabilistic method


In 1990 Bender, Canfield, and McKay gave an asymptotic formula for the number of connected graphs on [n] = {1, 2, . . . , n} with m edges, whenever n→∞ and (Formula presented.). We give an asymptotic formula for the number Cr(n,m) of connected r-uniform hypergraphs on [n] with m edges, whenever r ≥ 3 is fixed and m = m(n) with m/n→∞, that is, the average degree tends to infinity. This complements recent results of Behrisch, Coja-Oghlan, and Kang (the case m = n/(r − 1) + Θ(n)) and the present authors (the case m = n/(r − 1) + o(n), ie, “nullity” or “excess” o(n)). The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. The arguments are much simpler than in the sparse case; in particular, we can use “smoothing” techniques to directly prove the local limit theorem, without needing to first prove a central limit theorem.

Publication Title

Random Structures and Algorithms