On the representation number of grid graphs and cylindric grid graphs

TL;DR

This paper investigates the representation number of grid and cylindrical grid graphs, establishing that these graphs require three copies of each vertex for representation when dimensions are sufficiently large, and explores related toroidal grid graphs.

Contribution

It extends known results by determining the representation number for m x n grid and cylindrical grid graphs, and discusses conjectures related to toroidal grid graphs.

Findings

Grid graphs have representation number 3 for m ≥ 3.

Cylindrical grid graphs have representation number 3 for m ≥ 2.

Discussion of toroidal grid graphs and related conjecture.

Abstract

The representation number of a graph is the minimum number of copies of each vertex required to represent the graph as a word, such that the letters corresponding to vertices $x$ and $y$ alternate if and only if $xy$ is an edge in the graph. It is known that path graphs, circle graphs, and ladder graphs have representation number 2, while prism graphs have representation number 3. In this paper, we extend these results by showing that generalizations of the aforementioned graphs -- namely, the $m \times n$ grid graphs and $m \times n$ cylindrical grid graphs -- have representation number $3$ for $m \geq 3$ and $m \geq 2$, respectively, and $n\geq 3$. Furthermore, we discuss toroidal grid graphs in the context of word-representability, which leads to an interesting conjecture.