Group distance magic cubic graphs

Abstract

A $\hat{I}“$\emph{-distance magic labeling} of a graph $G=(V,E)$ with $|V|=n$ is a bijection $\ell$ from $V$ to an Abelian group $\hat{I}“$ of order $n$, for which there exists $\hat{I}¼\in \hat{I}“$, such that the weight $w(x)={\sum }_{y\in N(x)}\ell (y)$ of every vertex $x\in V$ is equal to $\hat{I}¼$. In this case, the element $\hat{I}¼$ is called the \emph{magic constant of} $G$. A graph $G$ is called a \emph{group distance magic} if there exists a $\hat{I}“$-distance magic labeling of $G$ for every Abelian group $\hat{I}“$ of order $n$. In this paper, we focused on cubic $\hat{I}“$-distance magic graphs as well as some properties of such graphs.