Group labelings of graphs

Abstract Given a graph Γ an abelian group G, and a labeling of the vertices of Γ with elements of G, necessary and sufficient conditions are stated for the existence of a labeling of the edges in which the label of each vertex equals the product of the labels of its incident edges. Such an edge labeling is called compatible. For vertex labelings satisfying these conditions, the set of compatible edge labelings is enumerated.