Connectivity properties of random subgraphs of the cube


The n‐dimensional cube Qn 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 Qn is an n‐connected graph of diameter and radius n. Write M = n2n−1 = e(Qn) for the size of Qn. Let Q̃ = (Qt)M0 be a random Q̃‐process. Thus Qt is a spanning subgraph of Qn of size t, and Qt is obtained from Qt‐1 by the random addition of an edge of Qn not in Qt‐1. Let tk = 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 t1 the graph Qt becomes connected and that in fact the diameter of Qt at this point is n + 1. Here we generalize this result by showing that, for any fixed k≧2, almost surely at time tk 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 tk. © 1995 John Wiley & Sons, Inc. Copyright © 1995 Wiley Periodicals, Inc., A Wiley Company

Publication Title

Random Structures & Algorithms