Note on group distance magic complete bipartite graphs

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}\ell(y)$ of every vertex $x \in V$ is equal to the same element $\mu \in \Gamma$, called the \emph{magic constant}. A graph $G$ is called a \emph{group distance magic graph} if there exists a $\Gamma$-distance magic labeling for every Abelian group $\Gamma$ of order $|V(G)|$. In this paper we prove that some complete $k$-partite graphs are $\mathbb{Z}_p$-distance magic. Moreover we prove that $K_{m,n}$ is a group distance magic if and only if $n+m \not \equiv 2 \pmod 4$. We also show that if $n+m \equiv 2 \pmod 4$, then there does not exist a group $\Gamma$ of order $n+m$ such that there exists a $\Gamma$-distance labeling for $K_{m,n}$.