On the strength of connectedness of a random hypergraph

Abstract

Bollobás and Thomason (1985) proved that for each k = k(n) ∈ [1; n - 1], with high probability, the random graph process, where edges are added to vertex set V = [n] uniformly at random one after another, is such that the stopping time of having minimal degree k is equal to the stopping time of becoming k-(vertex-)connected. We extend this result to the d-uniform random hypergraph process, where k and d are fixed. Consequently, for m = n/d (ln n + (k - 1) ln ln n + c) and, the probability that the random hypergraph models Hd(n;m) and Hd(n; p) are k-connected tends to e-e-c/(k-1)!: