Face-Magic Labelings of Some Grid-Related Graphs

A type (1, 1, 1) face-magic labeling of a planar graph G = (V, E, F) is a bijection from V ∪ E ∪ F to the set of labels {1, 2, . . . , |V | + |E| + |F|} such that the weight of every n-sided face of G is equal to the same fixed constant. The weight of a face F ∈ F is equal to the sum of the labels of the vertices, edges, and face that determine F. It is known that the grid graph Pm□Pn admits a type (1, 1, 1) facemagic labeling, but the proof in the literature is quite lengthy. We give a simple proof of this result and show two more infinite families of gridded graphs admit type (1, 1, 1) face-magic labelings.