Assorted Musings on Dimension-critical Graphs

Abstract

For a finite simple graph $G$, say $G$ is of dimension $n$, and write $\dim (G)=n$, if $n$ is the smallest integer such that $G$ can be represented as a unit-distance graph in ${\mathbb{R}}^n$. Define $G$ to be \emph{dimension-critical} if every proper subgraph of $G$ has dimension less than $G$. In this article, we determine exactly which complete multipartite graphs are dimension-critical. It is then shown that for each $n\ge 2$, there is an arbitrarily large dimension-critical graph $G$ with $\dim (G)=n$. We then pose and expound upon a number of questions related to this subject matter, questions that hopefully will prompt future research.