Assorted Musings on Dimension-critical Graphs

TL;DR

This paper characterizes dimension-critical complete multipartite graphs and demonstrates the existence of arbitrarily large such graphs for each dimension n, opening avenues for further research in graph dimension theory.

Contribution

It precisely identifies which complete multipartite graphs are dimension-critical and proves the existence of arbitrarily large dimension-critical graphs for each dimension n.

Findings

Complete multipartite graphs that are dimension-critical are fully characterized.

For each dimension n ≥ 2, arbitrarily large dimension-critical graphs with that dimension exist.

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 $n \geq 2$, 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.