Note on group distance magic graphs $G[C_4]$

A \emph{group distance magic labeling} or a $\gr$-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group $\gr$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element $μ\in \gr$, called the magic constant. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that $°(v)\equiv c \imod {2^{p+1}}$ for some constant $c$ for any $v\in V(G)$, then there exists an $\gr$-distance magic labeling for any Abelian group $\gr$ for the graph $G[C_4]$. Moreover we prove that if $\gr$ is an arbitrary Abelian group of order $4n$ such that $\gr \cong \zet_2 \times\zet_2 \times \gA$ for some Abelian group $\gA$ of order $n$, then exists a $\gr$-distance magic labeling for any graph $G[C_4]$.

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