Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids

TL;DR

This paper proves that in four dimensions, prismatoids cannot have a width greater than their dimension, by analyzing graph embeddings on a sphere derived from the normal fans of their bases.

Contribution

It establishes that 4-dimensional prismatoids cannot exceed width 4, resolving a key case related to the Hirsch conjecture and extending previous results in dimension five.

Findings

4-dimensional prismatoids have width at most 4

Graph embeddings on a sphere are used to analyze prismatoid properties

Supports the Hirsch conjecture in dimension four

Abstract

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the existence of counter-examples to the Hirsch conjecture is equivalent to that of $d$-prismatoids of width larger than $d$, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases.