# Cube-magic labelings of grids

## TL;DR

This paper proves that vertices and edges of multi-dimensional grid graphs can be labeled to ensure all subcubes have the same sum, establishing a supermagic property for these grids.

## Contribution

It introduces a new labeling method for grid graphs that guarantees uniform sums over all subcubes, extending supermagic labelings to higher-dimensional grids.

## Key findings

- Vertices and edges of grid graphs can be labeled to achieve supermagic sums.
- All subgraphs isomorphic to a d-cube have the same total label sum.
- The method applies to any dimension d ≥ 2.

## Abstract

We show that the vertices and edges of a $d$-dimensional grid graph $G=(V,E)$ ($d\geqslant 2$) can be labeled with the integers from $\{1,\ldots,\lvert V\rvert\}$ and $\{1,\ldots,\lvert E\rvert\}$, respectively, in such a way that for every subgraph $H$ isomorphic to a $d$-cube the sum of all the labels of $H$ is the same. As a consequence, for every $d\geqslant 2$, every $d$-dimensional grid graph is $Q_d$-supermagic where $Q_d$ is the $d$-cube.