Spanning subgraphs of Random Graphs (A research problem)

We propose a problem concerning the determination of the threshold function for the edge probability that guarantees, almost surely, the existence of various sparse spanning subgraphs in random graphs. We prove some bounds and demonstrate them in the cases of a d-cube and a two dimensional lattice. B. Bollobás (cf. e.g., [3]) raised the following problem: Let G be a random graph with n = 2d vertices, in which each edge is taken randomly and independently with probability p = 1− ε, where ε is a positive small constant. Is it true that for d > d(ε) almost surely G contains a copy of the d-cube, Qd? Note that Qd has 2d−1d = O(n log n) edges, and is thus a relatively sparse graph. Here we show that the answer is ”yes” for every fixed p > 1/2 and observe that it is ”no” for p ≤ 1/4. This is a special case of the following general theorem.