RADIUS AND DIAMETER OF RANDOM SUBGRAPHS OF THE HYPERCUBE

Let G(n) = (V(n), E(n)) be the random subgraph of the n-cube graph Q(n) produced as follows: V(n) is randomly sampled from the vertex set of Q(n) so that P{x is-an-element-of V(n)} = p(v) independently for each vertex x is-an-element-to Q(n). Then E(n) is randomly sampled from the set of edges induced by V(n) in Q(n) so that P{(x, y) is-an-element-of E(n)) = p(e) independently for each induced edge (x, y). Let p(v) and p(e) be fixed probabilities such that 0 < p = p(v)p(e) less-than-or-equal-to 1 and m(p) = [-1/log2(1 - p)]. We show that for the radius R(G(n)) and the diameter D(G(n)) of the main component of G(n) almost surely the following inequalities hold n + m(p) less-than-or-equal-to D(G(n)) less-than-or-equal-to n + m(p) + 8, n - 2 less-than-or-equal-to R(G(n)) less-than-or-equal-to n.