Simple polytopes without small separators, II: Thurston's bound

TL;DR

This paper constructs specific 4-dimensional simple polytopes with large minimal separators, confirming Thurston's claim and revealing new structural properties of such polytopes' graphs.

Contribution

It provides a construction of simple 4-polytopes with large separators, resolving a previously lost proof of Thurston's claim.

Findings

Graphs of the constructed polytopes have separators of size at least Ω(n/ log n)

The polytopes are formed by cutting off vertices and edges of neighborly cubical polytopes

The resulting graphs are 4-regular and contain 3-regular cube-connected cycle graphs as minors

Abstract

We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\Omega(n/\log n)$. This establishes a strong form of a claim by Thurston, for which the construction and proof had been lost. We construct the polytopes by cutting off the vertices and then the edges of a particular type of neighborly cubical polytopes. The graphs of simple polytopes thus obtained are 4-regular; they contain 3-regular "cube-connected cycle graphs" as minors of spanning subgraphs.