On the Euclidean dimension of 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.