Perfect matchings in grid graphs after vertex deletions

We investigate the d-dimensional grid graph [m]x[m]x...x[m] for even m. The graph is bipartite. If we choose a subset B' of the black vertices and a subset W' of the white vertices then the graph obtained by deleting B' and W' has a perfect matching if it satisfies the following conditions 1)|B'|=|W'| 2) each pair x,y in B' are at distance at least c.m^{1/d} 3) each pair x,y in W' are at distance at least c.m^{1/d} A value for c is given as a function of d. The factor m^{1/d} is in some sense best possible.