For a planar graph G of order n, let F (G) be the set of all faces of G embedded into R 2, including the exterior face. A bijective vertex labeling f: V (G) → 1, 2,. . . , n induces a face labeling f ∗: F (G) → N defined by setting f ∗ (F) equal to the sum of all labels of the boundary vertices of F. The graph G is said to be hyper face-magic if there exists a vertex labeling whose induced face labeling is constant. In this paper, we state properties of hyper face-magic graphs and construct various classes of hyper face-magic graphs.
Belgram et al. (Thu,) studied this question.