TY - JOUR
AU - Luczak, T.
AB - 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 t(k) = 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(1) 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 t(k) 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(k). © 1995 John Wiley & Sons, Inc.
TI - Connectivity properties of random subgraphs of the cube
JF - Random Structures and Algorithms
DO - 10.1002/rsa.3240060210
DA - 1995-03-01
UR - https://www.deepdyve.com/lp/wiley/connectivity-properties-of-random-subgraphs-of-the-cube-WflWMoYY34
SP - 221
EP - 230
VL - 6
IS - 2‐3
DP - DeepDyve
ER -