Connectivity Properties of Random Subgraphs of the Cube

The n-dimensional cube Q is the graph whose vertices are the subsets of {1, . . . , n} where two such vertices are adjacent if and only if their symmetric difference is a singleton. Clearly Q is an n-connected graph of diameter and radius n. Write M = n2n−1 = e(Q) for the size of Q. Let Q̃ = (Qt) M 0 be a random Q -process. Thus Qt is a spanning subgraph of Q n of size t, and Qt is obtained from Qt−1 by the random addition of an edge of Q n not in Qt−1. Let t = τ(Q̃; δ ≥ k) be the hitting time of the property of having minimal degree at least k. It is shown in [5] that, almost surely, at time t the graph Qt becomes connected and that in fact the diameter of Qt at this point is n+ 1. Here we generalise this result by showing that, for any fixed k ≥ 2, almost surely at time t the graph Qt acquires the extremely strong property that any two of its vertices are connected by k internally vertex-disjoint paths each of length at most n, except for possibly one, which may have length n + 1. In particular, the hitting time of k-connectedness is almost surely t.