Grid-magic labelings of grid unions

Abstract

Consider G and H as simple graphs. A graph G has an H-covering if every edge of G belongs to a subgraph of G which the subgraph is isomorphic to H. The graph G is said to be H-magic if there exists a total labeling of G such that for every subgraph H′ of G where H′ is isomorphic to H, then the sum of all labels in H′ is constant. In this paper, we study grid-like graphs G, whose faces in F(G) are labeled. The magic labeling of type (1,1,1) means that the label is assigned to all vertices, edges, and faces, and the weight is calculated as -v-V(G) l(v)+-e-E(G) l(e)+-f-F(G) l(f) with all weight for every face being constant. We show that certain unions of grid graphs are H-magic, where H is a grid graph.