Simple polytopes without small separators

TL;DR

This paper constructs specific simple 4-dimensional polytopes with large graph separators, disproving a longstanding conjecture by Kalai from 1991/2004.

Contribution

It introduces a method to create simple polytopes with large separators, challenging previous assumptions about their size.

Findings

Constructed simple 4-polytopes with separators of size at least Ω(n/ log^{3/2} n)

Disproved Kalai's conjecture from 1991/2004

Demonstrated new properties of neighborly cubical polytopes

Abstract

We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\Omega(n/\log^{3/2}n)$. This disproves a conjecture by Kalai from 1991/2004.