The component structure of dense random subgraphs of the hypercube

Given $p \in (0,1)$, we let $Q_p= Q_p^d$ be the random subgraph of the $d$-dimensional hypercube $Q^d$ where edges are present independently with probability $p$. It is well known that, as $d \rightarrow \infty$, if $p>\frac12$ then with high probability $Q_p$ is connected; and if $p<\frac12$ then with high probability $Q_p$ consists of one giant component together with many smaller components which form the `fragment'. Here we fix $p \in (0,\frac12)$, and investigate the fragment, and how it sits inside the hypercube. In particular we give asymptotic estimates for the mean numbers of components in the fragment of each size, and describe their asymptotic distributions and indeed their joint distribution, much extending earlier work of Weber

Connectivity properties of random subgraphs of the cube

The n-dimensional cube Q n 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 Q n is an n-connected graph of diameter and radius n. Write M = n2 n−1 = e(Q n) for the size of Q n. Let ˜Q = (Qt) M 0 be a random Q n-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 n 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. 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 generalise 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)