The number of connected sparsely edged uniform hypergraphs
Discrete Mathematics
Growth of components in random graphs
Proceedings of the ninth international conference on on Random structures and algorithms
The phase transition in a random hypergraph
Journal of Computational and Applied Mathematics - Special issue: Probabilistic methods in combinatorics and combinatorial optimization
Graphs and Hypergraphs
Generatingfunctionology
Analytic Combinatorics
Multicyclic components in a random graph process
Random Structures & Algorithms
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
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Denote by an ℓ-component a connected b-uniform hypergraph with k edges and k(b–1) – ℓ vertices. We prove that the expected number of creations of ℓ-component during a random hypergraph process tends to 1 as ℓ and b tend to ∞ with the total number of vertices n such that $\ell = o\left( \sqrt[3]{\frac{n}{b}} \right)$. Under the same conditions, we also show that the expected number of vertices that ever belong to an ℓ-component is approximately 121/3 (b–1)1/3 ℓ1/3n2/3. As an immediate consequence, it follows that with high probability the largest ℓ-component during the process is of size O( (b–1)1/3 ℓ1/3n2/3 ). Our results give insight about the size of giant components inside the phase transition of random hypergraphs.