Face numbers of centrally symmetric polytopes from split graphs

TL;DR

This paper studies Hansen polytopes derived from split graphs, confirming a conjecture about their face counts, identifying special subclasses, and introducing a new family with specific face properties.

Contribution

It proves Kalai's 3^d-conjecture for Hansen polytopes of split graphs and characterizes Hanner polytopes within this class, also presenting a new family with exactly 3^d+16 faces.

Findings

Kalai's 3^d-conjecture holds for Hansen polytopes of split graphs

Hanner polytopes correspond to threshold graphs within this class

A new family of Hansen polytopes with 3^d+16 faces is identified

Abstract

We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes (they all have at least 3^d nonempty faces) and show that the Hanner polytopes among them (which have exactly 3^d nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only 3^d+16 nonempty faces.