Matching Extendability in Hypercubes

In a bipartite graph G, a set S V (G) is deficient if |N (S) |<|S|. A matching M (with vertex set U) is k-suitable if G-U has no deficient set of size less than k. Let fₖ (d) be the largest r such that in the d-dimensional hypercube Qd every k-suitable matching with at most r edges extends to a perfect matching. We generalize results of Limaye and Sarvate by proving that fₖ (d) =k (d-k) +k-12 for k d-3. To this end we prove lower bounds on the sizes of neighborhoods of vertex sets in Qd. We also prove that every induced matching in Qd extends to a perfect matching.