Distance magic graphs G×CnG×Cn

A ΓΓ-distance magic labeling of a graph G=(V,E)G=(V,E) with |V|=n|V|=n is a bijection ff from VV to an Abelian group ΓΓ of order nn such that the weight w(x)=∑y∈NG(x)f(y)w(x)=∑y∈NG(x)f(y) of every vertex x∈Vx∈V is equal to the same element μ∈Γμ∈Γ, called the magic constant.In this paper we will show that if GG is a graph of order n=2p(2k+1)n=2p(2k+1) for some natural numbers pp, kk such that deg(v)≡c(mod2p+2) for some constant cc for any v∈V(G)v∈V(G), then there exists a ΓΓ-distance magic labeling for any Abelian group ΓΓ of order 4n4n for the direct product G×C4G×C4. Moreover if cc is even, then there exists a ΓΓ-distance magic labeling for any Abelian group ΓΓ of order 8n8n for the direct product G×C8G×C8.