On the Diameter and Radius of Random Subgraphs of the Cube

The n-dimensional cube Q is the graph whose vertices are the subsets of {1, . . . , n}, with two vertices adjacent if and only if their symmetric difference is a singleton. Clearly Q has 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 not in Qt−1. Let t (k) = τ(Q̃; δ ≥ k) be the hitting time of the property of having minimal degree at least k. We show that the diameter dt = diam(Qt) of Qt in almost every Q̃ behaves as follows: dt starts infinite and is first finite at time t , it equals n + 1 for t ≤ t < t, and dt = n for t ≥ t. We also show that the radius of Qt is first finite for t = t, when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Qt for t = Ω(M).