On a problem of Specker about Euclidean representations of finite graphs

TL;DR

This paper discusses Euclidean representations of finite graphs, proving that any non-trivial finite graph can be embedded in a Euclidean space of dimension one less than the number of vertices, extending to edge-colored graphs.

Contribution

It provides a proof that non-complete, non-independent finite graphs are representable in a Euclidean space of dimension |G|-2, generalizing Specker's problem.

Findings

Finite graphs (not complete or independent) are representable in R^{|G|-2}.

The result extends to finite complete edge-colored graphs.

The proof confirms a conjecture by Einhorn and Schoenberg.

Abstract

Say that a graph $G$ is \emph{representable in $\R ^n$} if there is a map $f$ from its vertex set into the Euclidean space $\R ^n$ such that $\| f(x) - f(x')\| = \| f(y) - f(y')\|$ iff $\{x,x'\}$ and $\{y, y'\}$ are both edges or both non-edges in $G$. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg: if $G$ finite is neither complete nor independent, then it is representable in $\R ^{|G|-2}$. A similar result also holds in the case of finite complete edge-colored graphs.