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.