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

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

2011-02-13

arxiv.org/abs/1102.2645v2

Mathematics

Combinatorics

This paper and arXiv:1101.3050 have been merged, forming now the paper arXiv:1104.2630

Scientific paper

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 author recently showed in arXiv:1006.2814 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 $Q$.

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