Universal geometric non-embedding of random regular graphs

TL;DR

This paper proves that typical large random regular graphs cannot be embedded as geometric graphs into low-dimensional normed spaces, extending known results about expander graphs to a universal setting across all norms.

Contribution

It establishes a universal non-embeddability result for typical random regular graphs into low-dimensional normed spaces, independent of the specific norm used.

Findings

Random regular graphs cannot be embedded into low-dimensional normed spaces.

The non-embeddability holds with high probability for large graphs.

The proof uses a multiscale -net argument.

Abstract

Let $\Delta \ge 3$ be fixed, $n \ge n_\Delta$ be a large integer. It is a classical result that $\Delta$--regular expanders on $n$ vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than $c \log n$, for some universal constant $c$. We show that for typical $\Delta$-regular graphs, this obstruction is universal with respect to the choice of norm. More precisely, for a uniform random $\Delta$-regular graph $G$ on $n$ vertices, it holds with high probability: there is no normed space of dimension less than $c\log n$ which admits a geometric graph isomorphic to $G$. The proof is based on a seeded multiscale $\varepsilon$--net argument.