Creation and growth of components in a random hypergraph process

Abstract

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 ℓ = o (3√n/b). Under the same conditions, we also show that the expected number of vertices that ever belong to an ℓ-component is approximately 12 1/3(b-1)1/3ℓ1/3n2/3. As an immediate consequence, it follows that with high probability the largest ℓ-component during the process is of size O((b-1)1/3ℓ 1/3n2/3). Our results give insight about the size of giant components inside the phase transition of random hypergraphs. © Springer-Verlag Berlin Heidelberg 2006.