Distance magic labelings of product graphs

TL;DR

This paper investigates distance magic labelings in product graphs, exploring conditions and constructions using magic rectangles for Cartesian, strong, lexicographic, and Kronecker products.

Contribution

It provides new methods for constructing distance magic labelings of product graphs using magic rectangle sets and extends existing theory to various graph products.

Findings

Distance magic labelings exist for certain product graphs.

Magic rectangle sets are effective in constructing labelings.

Results extend the understanding of labelings in complex graph products.

Abstract

A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of $x$. In this paper we shall study distance magic labelings of graphs obtained from four graph products: cartesian, strong, lexicographic, and cronecker. We shall utilise magic rectangle sets and magic column rectangles to construct the labelings.