Distance magic labeling and two products of graphs

TL;DR

This paper studies distance magic labelings in graphs, introduces a new subclass, and explores their properties under graph products, including characterizations for specific cases like the direct product of cycles.

Contribution

It introduces a natural subclass of distance magic graphs and analyzes their closure properties under direct and lexicographic products, with characterizations for certain graph products.

Findings

The subclass is closed under direct product with regular graphs.

The subclass is closed as a second factor in lexicographic product with regular graphs.

Distance magic graphs among direct products of two cycles are characterized.

Abstract

Let $G=(V,E)$ be a graph of order $n$. A distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow {1,...,n}$ for which there exists a positive integer $k$ such that $\sum_{x\in N(v)}\ell (x)=k$ for all $v\in V $, where $N(v)$ is the neighborhood of $v$. We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.