Embedding graphs in Euclidean space

TL;DR

This paper investigates the Euclidean embedding dimension of graphs, establishing bounds based on edges and degree, and explores Ramsey-type results for graph embeddings in Euclidean space and spheres.

Contribution

It improves existing bounds on graph embedding dimensions and proves new Ramsey results for embeddings in Euclidean space and spheres.

Findings

Graphs with fewer than inom{d+2}{2} edges have dimension at most d

Graphs with maximum degree d have dimension at most d

In any red-blue edge coloring of K_{2d}, one color class can be embedded in -space

Abstract

The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances). Answering a question of Erd\H{o}s and Simonovits [Ars Combin. 9 (1980) 229--246], we show that any graph with less than $\binom{d+2}{2}$ edges has dimension at most $d$. Improving their result, we prove that that the dimension of a graph with maximum degree $d$ is at most $d$. We show the following Ramsey result: if each edge of the complete graph on $2d$ vertices is coloured red or blue, then either the red graph or the blue graph can be embedded in Euclidean $d$-space. We also derive analogous results for embeddings of graphs into the $(d-1)$-dimensional sphere of radius $1/\sqrt{2}$.