Counting Connected Graphs and Hypergraphs via the Probabilistic Method

While it is exponentially unlikely that a sparse random graph or hypergraph is connected, with probability 1 − o(1) such a graph has a “giant component” that, given its numbers of edges and vertices, is a uniformly distributed connected graph. This simple observation allows us to estimate the number of connected graphs, and more generally the number of connected d-uniform hypergraphs, on n vertices with ((d − 1)−1 + ε)n ≤ m = o(n ln n) edges, where ε > 0 is arbitrarily small but independent of n. We also estimate the probability that a binomial random hypergraph Hd(n, p) is connected, and determine the expected number of edges of Hd(n, p) given that it is connected. This extends prior work of Bender, Canfield, and McKay [5] on the number of connected graphs. While [5] is based on a recursion relation satisfied by the number of connected graphs, so that the argument is to some extent enumerative, we present a purely probabilistic approach.

• Type: Conference
• Year: 2004
• Where: APPROX
• Authors: Amin Coja-Oghlan, Cristopher Moore, Vishal Sanwalani
• Keywords: Algorithms, APPROX 2004, Connected D-uniform Hypergraphs, Connected Graphs, Sparse Random Graph