On a problem of Specker about Euclidean representations of finite graphs

Details On a problem of Specker about Euclidean representations of finite graphs On a problem of Specker about Euclidean representations of finite graphs

2008-10-14

arxiv.org/abs/0810.2359v2

Mathematics

Combinatorics

6 pages, 2 figures. The previous version of this article will not be published because the question it answers was actually an

Scientific paper

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.

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