Note on group distance magic complete bipartite graphs

TL;DR

This paper investigates group distance magic labelings of complete bipartite graphs and characterizes when such labelings exist based on the group order and graph parameters, extending understanding of magic labelings in graph theory.

Contribution

It proves that some complete k-partite graphs are -distance magic and characterizes the conditions under which complete bipartite graphs are group distance magic.

Findings

$K_{m,n}$ is group distance magic iff $m+n ot\equiv 2 \pmod 4$

Some complete $k$-partite graphs are -distance magic

No group -distance magic labelings exist for $K_{m,n}$ when $m+n \equiv 2 \pmod 4$

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 $n+m \not \equiv 2 \pmod 4$. 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 $K_{m,n}$.