Counting 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 the nullity m-n+1 tend to infinity. Let C r(n,t) be the number of connected r-uniform hypergraphs on [n] with nullity t = (r-1)m-n+1, where m is the number of edges. For r ≥ 3, asymptotic formulae for C r(n,t) are known only for partial ranges of the parameters: in 1997 Karoński and Łuczak gave one for t = o(log n/log log n), and recently Behrisch, Coja-Oghlan and Kang gave one for t=Θ(n). Here we prove such a formula for any fixed r ≥ 3 and any t = t(n) satisfying t = o(n) and t→ as n→, complementing the last result. This leaves open only the case t/n→, which we expect to be much simpler, and will consider in future work. 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. We deduce this from the corresponding central limit theorem by smoothing techniques.

Publication Title

Combinatorics Probability and Computing