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arXiv · 2106.05333

Assorted Musings on Dimension-critical Graphs

Matt Noble
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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.

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Matt Noble. 2021-06-09. Assorted Musings on Dimension-critical Graphs. https://arxiv.org/abs/2106.05333

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