Note on group distance magicness on product graphs

Abstract

If $l$ is a bijection from the vertex set $V(G)$ of a graph $G$ to an additive abelian group $\Gamma$ of $|V(G)|$ elements such that for any vertex $u$ of $G$, the weight $\sum_{v\in N_{G}(u)}l(v)$ is $\mu$, where $\mu \in \Gamma$, then $l$ is a $\Gamma$-distance magic labeling of $G$. A graph $G$ that admits such an $l$ is $\Gamma$-distance magic and if $G$ is $\Gamma$-distance magic for every such $\Gamma$, then $G$ is a group distance magic graph.

In this paper, we provide some results on the group distance magicness of the lexicographic and direct product of two graphs. By proving a few necessary conditions, we characterize the group distance magicness of a tree. In addition, we find three techniques to construct group distance magic graphs recursively from the existing ones and with respect to any abelian group with one involution, we determine infinitely many nongroup distance magic graphs.