Characterization of completely k-magic regular graphs

Let k ∈ ℕ and c ∈ ℤk. A graph G is said to be c-sum k-magic if there is a labeling ℓ : E(G) → ℤk \ {0} such that Σu∈N(v) ℓ(uv) ≡ c (mod k) for every vertex v of G, where N(v) is the neighborhood of v in G. We say that G is completely k-magic whenever it is c-sum k-magic for every c ∈ ℤk. In this paper, we characterize all completely k-magic regular graphs.