Assorted Musings on Dimension-critical Graphs [article]

Matt Noble
2023 arXiv   pre-print
For a finite simple graph G, say G is of dimension n, and write (G) = n, if n is the smallest integer such that G can be represented as a unit-distance graph in ℝ^n. Define G to be 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 ≥ 2, there is an arbitrarily large dimension-critical graph G with (G) = n. We then pose and expound upon a
more » ... r of questions related to this subject matter, questions that hopefully will prompt future research.
arXiv:2106.05333v2 fatcat:id7mk7h3hnconfqv2jno5eq3iq