Zero-sum magic and null sets of planar graphs

For any h ∈ N, a graph G =(V, E) is said to be h-magic if there exists a labeling l : E(G) → Zh - {0} such that the induced vertex labeling l+: V(G)→ Zh defined by l+(v) = Σ uv∈E(G) l(uv) is a constant map. When this constant is 0 we call G a zero-sum h-magic graph. The null set of G is the set of all natural numbers h ∈ N for which G admits a zero-sum h-magic labeling. A graph G is said to be uniformly null if every magic labeling of G induces zero sum. In this paper we will identify the null sets of certain planar graphs such as wheels and fans.