Large Matchings and nearly Spanning, nearly Regular Subgraphs of Random Subgraphs

Given a graph G and p ∈ [0, 1], the random subgraph G_p is obtained by retaining each edge of G independently with probability p. We show that for every ɛ > 0, there exists a constant C > 0 such that the following holds. Let d ≥ C be an integer, let G be a d-regular graph and let p ≥^Cd . Then, with probability tending to one as |V (G)| tends to infinity, there exists a matching in G_p covering at least (1 − ɛ)|V (G)| vertices. We further show that for a wide family of d-regular graphs G, which includes the d-dimensional hypercube, for any p ≥^log5 d d with probability tending to one as d tends to infinity, G_p contains an induced subgraph on at least (1 − o(1))|V (G)| vertices, whose degrees are tightly concentrated around the expected average degree dp.