Connectivity of the k-Out Hypercube

Abstract

In this paper we study the connectivity properties of the random subgraph of the ncube generated by the k-out model and denoted by Q n (k). Let k be an integer, 1 ≤ k ≤ n−1. We let Q n (k) be the graph that is generated by independently including for every v ∈ V (Q n ) a set of k distinct edges chosen uniformly from all the n k sets of distinct edges that are incident to v. We study connectivity the properties of Q n (k) as k varies. We show that w.h.p. 1 Q n (1) does not contain a giant component i.e. a component that spans Ω(2 n ) vertices. Thereafter we show that such a component emerges when k = 2. In addition the giant component spans all but o(2 n ) vertices and hence it is unique. We then establish the connectivity threshold found at k 0 = log 2 n − 2 log 2 log 2 n. The threshold is sharp in the sense that Q n (⌊k 0 ⌋) is disconnected but Q n (⌈k 0 ⌉ + 1) is connected w.h.p. Furthermore we show that w.h.p. Q n (k) is k-connected for every k ≥ ⌈k 0 ⌉ + 1. * email:manastos@andrew.cmu.edu 1 we say that a sequence of events {E n } holds with high probability (w.h.p.) or equivalency almost surely if P(E n ) → 1 as n → ∞.