Note on group distance magic complete bipartite graphs

Abstract

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 $n+m\not\equiv 2(\mathrm{mod}4)$. We also show that if $n+m\equiv 2(\mathrm{mod}4)$, then there does not exist a group $\Gamma$ of order $n+m$ such that there exists a $\Gamma$-distance labeling for $K_{m,n}$.

Since the politc of the Journal I submitted the paper I need to withdraw the paper from arxiv

References in corpus (1)

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

Cited by in corpus (1)

• Zero-sum partitions of Abelian groups of order $2^n$