Let G, = (Vn, En) be the random subgraph of the n-cube graph Q" produced as follows: V,, is randomly sampled from the vertex set of Q" so that P { x E V,,} = p , independently for each vertex x E Q". Then En is randomly sampled from the set of edges induced by V,, in Q" so that P { ( x , y) E En} = p , independently for each induced edge (x, y). Let p , and p , be fixed probabilities such that 0 < p = p u p , 5 1 and mp = 1-1 /log,( 1 -p ) ] . We show that for the radius R(G,) and the diameter D(G,) of the main component of G, almost surely the following inequalities hold n + mp I D(G,)I n + m p + 8 , n -2 5 R( G,, ) I n .
We consider two types of random subgraphs of the n-cube Q, obtained by independent deletion the vertices (together with all edges incident with them) or the edges of Q,,, respectively, with a prescribed probability q = 1 -p . For these two probabilistic models we determine some values of the probability p for which the number of (isolated) L-dimensional subcubes or the number of vertices of a given degree k, respectively, has asymptotically a Poisson or a Normal distribntion. The technique which will he wed is that of Poisson convergence introduced by BARBOUR [I] (see also [a]).
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