Matching extendability in Cartesian products of cycles

In a bipartite graph G, a set (Formula presented) is deficient if |N(S)| \u3c |S|. A matching M with vertex set U is k-suitable if G − U has no deficient set of size less than k. Define the extremal function fk (G) to be the largest integer r such that every k-suitable matching in G with at most r edges extends to a perfect matching. Let G(2m)d be the d-fold Cartesian product of the cycle C2m,wherem ≥ 2. We extend results of Vandenbussche and West by showing that for any integers k and d such that (Formula presented), except when m =2 and d =1