On the diameter and radius of randon subgraphs of the cube


The n‐dimensional cube Qn 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 Qn has diameter and radious n. Write M = n2n‐1 = e(Qn) for the size of Qn. Let Q = (Qt)oM be a random Qn‐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 t(k) = τ(Q;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 t1, it equals n + 1 for t1 ⩽ t2 and dt, = n for t ⩾ t2. We also show that the radius of Qt, is first finite for t = t1, 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). © 1994 John Wiley & Sons, Inc. Copyright © 1994 Wiley Periodicals, Inc., A Wiley Company

Publication Title

Random Structures & Algorithms