Random Structures & Algorithms
The number of connected sparsely edged uniform hypergraphs
Discrete Mathematics
The phase transition in a random hypergraph
Journal of Computational and Applied Mathematics - Special issue: Probabilistic methods in combinatorics and combinatorial optimization
On the Fluctuations of the Giant Component
Combinatorics, Probability and Computing
Counting connected graphs inside-out
Journal of Combinatorial Theory Series B
Counting connected graphs asymptotically
European Journal of Combinatorics - Special issue on extremal and probabilistic combinatorics
Local Limit Theorems for the Giant Component of Random Hypergraphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
The order of the giant component of random hypergraphs
Random Structures & Algorithms
Birth and growth of multicyclic components in random hypergraphs
Theoretical Computer Science
Propagation connectivity of random hypergraphs
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Efficient sketches for the set query problem
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Asymptotic normality of the size of the giant component in a random hypergraph
Random Structures & Algorithms
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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(nlnn) 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 et al. (Random Struct Algorithm 1 (1990), 127–169) on the number of connected graphs. While Bender et al. (1990) 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. © 2007 Wiley Periodicals, Inc. Random Struct., 2007