Cube-magic labelings of grids

The graphs considered in this paper are finite, undirected and simple. For a graph G, we denote its vertex set by V (G) and its edge set by E(G). A graph labeling, as introduced in [8], is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Over the years, a large variety of different types of graph labelings have been studied, see [2] for an extensive survey. For a graph H, we say that a graph G admits an H-covering if every edge of G belongs to at least one subgraph of G which is isomorphic to H. A graph G = (V,E) which admits an H-covering is called H-magic if there exists a bijection F : V ∪E → {1, 2, . . . , |V |+ |E|} and a constant c = c(F ), which we call the H-magic sum of F , such that