Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or aG-distance magic labeling of a graph G = (V, E) with |V | = n is a bijection f from V to an Abelian group G of order n such that the weight w(x) = ∑y∈NG (x) f (y) of every vertex x ∈ V is equal to the same element μ ∈ G, called the magic constant. In this paper we will show that if G is a graph of order n = 2p(2k + 1) for some natural numbers p, k such that deg(v) ≡ c (mod 2p+1) for some constant c for any v ∈ V (G), then there exists a G-distance magic labeling for any Abelian group G of order 4n for the composition G[C4]. Moreover we prove that if G is an arbitrary Abelian group of order 4n such that G ∼= Z2 × Z2 × A for some Abelian group A of order n, then there exists a G-distance magic labeling for any graph G[C4], where G is a graph of order n and n is an arbitrary natural number.