On the products of group-magic graphs

Let A be an abelian group. We call a graph G = (V,E) A–magic if there exists a labeling f : E(G) → A− {0} such that the induced vertex set labeling f+ : V (G) → A, defined by f+(v) = Σf(u, v) where the sum is over all (u, v) ∈ E(G), is a constant map. For four classical products, we examine the A–magic property of the resulting graph obtained from the product of two A–magic graphs.