Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids
Combinatorics
2012-02-28 v1 Computational Geometry
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 -prismatoids of width larger than , 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.
Cite
@article{arxiv.1104.2630,
title = {Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids},
author = {Francisco Santos and Tamon Stephen and Hugh Thomas},
journal= {arXiv preprint arXiv:1104.2630},
year = {2012}
}
Comments
This paper merges and supersedes the papers arXiv:1101.3050 (of the last two authors) and arXiv:1102.2645 (of the first author)