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

We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\mathrm{\Omega }(n\mathrm{/}\log n)$. 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.