Note on group distance magic complete bipartite graphs

A $\mathrm{\Gamma }$-distance magic labeling of a graph $G=(V,E)$ with $\mathrm{\mid }V\mathrm{\mid }=n$ is a bijection $\mathrm{\ell }$ from $V$ to an Abelian group $\mathrm{\Gamma }$ of order $n$ such that the weight $w(x)={\sum }_{y\in N_G(x)}\mathrm{\ell }(y)$ of every vertex $x\in V$ is equal to the same element $\mu \in \mathrm{\Gamma }$, called the \emph{magic constant}. A graph $G$ is called a \emph{group distance magic graph} if there exists a $\mathrm{\Gamma }$-distance magic labeling for every Abelian group $\mathrm{\Gamma }$ of order $\mathrm{\mid }V(G)\mathrm{\mid }$. 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 $n+m\equiv ̸2(\mathrm{m}\mathrm{o}\mathrm{d}4)$. We also show that if $n+m\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, then there does not exist a group $\mathrm{\Gamma }$ of order $n+m$ such that there exists a $\mathrm{\Gamma }$-distance labeling for $K_{m,n}$.