Cube-magic Labelings Of Grids

We show that the vertices and edges of a d-dimensional grid graph G = (V, E) (d > 2) can be labeled with the integers from {1, . . . , |V |} and {1, . . . , |E|}, respectively, in such a way that for every subgraph H isomorphic to a d-cube the sum of all the labels of H is the same. As a consequence, for every d > 2, every d-dimensional grid graph is Qd-supermagic where Qd is the d-cube.