On the Euclidean dimension of graphs

TL;DR

This paper determines the Euclidean dimension of twelve well-known graphs, showing which can be embedded in the plane or in three-dimensional space, and provides explicit embeddings for each.

Contribution

It identifies the Euclidean dimension for twelve notable graphs and supplies explicit embeddings, advancing understanding of graph geometric representations.

Findings

Five graphs embed in the plane.

Seven graphs require three-dimensional space.

Explicit embeddings are provided for all graphs.

Abstract

The Euclidean dimension a graph $G$ is defined to be the smallest integer $d$ such that the vertices of $G$ can be located in $\mathbb{R}^d$ in such a way that two vertices are unit distance apart if and only if they are adjacent in $G$. In this paper we determine the Euclidean dimension for twelve well known graphs. Five of these graphs, D\"{u}rer, Franklin, Desargues, Heawood and Tietze can be embedded in the plane, while the remaining graphs, Chv\'{a}tal, Goldner-Harrary, Herschel, Fritsch, Gr\"{o}tzsch, Hoffman and Soifer have Euclidean dimension $3$. We also present explicit embeddings for all these graphs.