# Counting Connected Graphs and Hypergraphs via the Probabilistic Method

• Amin Coja-Oghlan
• Cristopher Moore
• Vishal Sanwalani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3122)

## Abstract

It is exponentially unlikely that a sparse random graph or hypergraph is connected, but such graphs occur commonly as the giant components of larger random graphs. 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 m=O(n) edges. We also estimate the probability that a binomial random hypergraph H d (n,p) is connected, and determine the expected number of edges of H d (n,p) conditioned on its being connected. This generalizes prior work of Bender, Canfield, and McKay [2] on the number of connected graphs; however, our approach relies on elementary probabilistic methods, extending an approach of O’Connell, rather than using powerful tools from enumerative combinatorics. We also estimate the probability for each t that, given k=O(n) balls in n bins, every bin is occupied by at least t balls.

## References

1. 1.
Barraez, D., Boucheron, S., Fernandez de la Vega, W.: On the fluctuations of the giant component. Combinatorics, Probability and Computing 9, 287–304 (2000)
2. 2.
Bender, E.A., Canfield, E.R., McKay, B.D.: The asymptotic number of labeled connected graphs with a given number of vertices and edges. Random Structures & Algorithms 1, 127–169 (1990)
3. 3.
Bender, E.A., Canfield, E.R., McKay, B.D.: Asymptotic properties of labeled connected graphs. Random Structures & Algorithms 3, 183–202 (1992)
4. 4.
Engel, A., Monasson, R., Hartmann, A.: On large deviation properties of Erdös- Rényi random graphs. Preprint rXiv, cond-mat/0311535 (2003) Google Scholar
5. 5.
Feller, W.: Introduction to probability theory and its applications. Wiley, Chichester (1968)
6. 6.
Janson, S., Luczak, T., Ruciński, A.: Random Graphs. Wiley, Chichester (2000)
7. 7.
Karonski, M., Luczak, T.: The phase transition in a random hypergraph. J. Comput. Appl. Math. 142, 125–135 (2002)
8. 8.
Karonski, M., Luczak, T.: The number of connected sparsely edged uniform hypergraphs. Discrete Math. 171, 153–168 (1997)
9. 9.
Luczak, T.: On the number of sparse connected graphs. Random Structures & Algorithms 1, 171–173 (1990)
10. 10.
Moser, L., Wyman, M.: Stirling numbers of the second kind. Duke Mathematical Journal 25, 29–43 (1958)
11. 11.
O’Connell, N.: Some large deviation results for sparse random graphs. Prob. Th. Relat. Fields 110, 277–285 (1998)
12. 12.
Pittel, B., Wormald, N.C.: Counting connected graphs inside out. J. Combinatorial Theory Series B (to appear) Google Scholar
13. 13.
Schmidt-Pruzan, J., Shamir, E.: Component structure in the evolution of random hypergraphs. Combinatorica 5, 81–94 (1985)
14. 14.
Temme, N.M.: Asymptotic estimates of Stirling numbers. Studies in Applied Mathematics 89, 233–243 (1993)

## Authors and Affiliations

• Amin Coja-Oghlan
• 1
• Cristopher Moore
• 2
• Vishal Sanwalani
• 2
1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany
2. 2.University of New MexicoAlbuquerqueUSA