Embedding a Pair of Graphs in a Surface, and the Width of 4-dimensional Prismatoids

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 of Q.

Authors and Affiliations

1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Av. de los Castros 48, 39005, Santander, Spain

   Francisco Santos

2. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada

   Tamon Stephen

3. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada

   Hugh Thomas

Corresponding author

Correspondence to Tamon Stephen.

Reprints and permissions