Birth and growth of multicyclic components in random hypergraphs

Abstract

Define an ℓ -component to be a connected b -uniform hypergraph with k edges and k ( b − 1 ) − ℓ vertices. In this paper, we investigate the growth of size and complexity of connected components of a random hypergraph process. We prove that the expected number of creations of ℓ -components during a random hypergraph process tends to 1 as b is fixed and ℓ tends to infinity with the total number of vertices n while remaining ℓ = o ( n 1 / 3 ) . We also show that the expected number of vertices that ever belong to an ℓ -component is ∼ 1 2 1 / 3 ℓ 1 / 3 n 2 / 3 ( b − 1 ) − 1 / 3 . We prove that the expected number of times hypertrees are swallowed by ℓ -components is ∼ 2 1 / 3 3 − 1 / 3 n 1 / 3 ℓ − 1 / 3 ( b − 1 ) − 5 / 3 . It follows that with high probability the largest ℓ -component during the process is of size of order O ( ℓ 1 / 3 n 2 / 3 ( b − 1 ) − 1 / 3 ) . Our results give insight into the size of giant components inside the phase transition of random hypergraphs and generalize previous results about graphs.