Ring-magic labelings of graphs

In this paper, a generalization of a group-magic graph is introduced and studied. Let R be a commutative ring with unity 1. A graph G = (V,E) is called R-ring-magic if there exists a labeling f : E → R−{0} such that the induced vertex labelings f : V → R, defined by f(v) = Σf(u, v) where (u, v) ∈ E, and f : V → R, defined by f(v) = Πf(u, v) where (u, v) ∈ E, are constant maps. General algebraic results for R-ringmagic graphs are established. In addition, Zn-ring-magic graphs and, in particular, trees are examined.