Group distance magic labeling of direct product of graphs

Abstract

Let G = (V, E) be a graph and Γ  an Abelian group, both of order n. A group distance magic labeling of G is a bijection ℓ: V → Γ  for which there exists μ ∈ Γ  such that ∑ _{x ∈ N(v)}ℓ(x) = μ for all v ∈ V, where N(v) is the neighborhood of v. In this paper we consider group distance magic labelings of direct product of graphs. We show that if G is an r -regular graph of order n and m = 4 or m = 8 and r is even, then the direct product C_m × G is Γ -distance magic for every Abelian group of order mn. We also prove that C_m × C_n is Z_mn-distance magic if and only if m ∈ {4, 8} or n ∈ {4, 8} or m, n ≡ 0 mod 4. It is also shown that if m, n not≡ 0 mod 4 then C_m × C_n is not Γ -distance magic for any Abelian group Γ  of order mn.