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

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 of Q.

References

  1. 1.

    Matschke, B., Santos, F., Weibel, C.: The width of 5-prismatoids and smaller non-Hirsch polytopes (in preparation)

  2. 2.

    Santos, F.: A counter-example to the Hirsch Conjecture. Preprint arXiv:1006.2814, June 2010

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Author information

Correspondence to Tamon Stephen.

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Santos, F., Stephen, T. & Thomas, H. Embedding a Pair of Graphs in a Surface, and the Width of 4-dimensional Prismatoids. Discrete Comput Geom 47, 569–576 (2012). https://doi.org/10.1007/s00454-011-9361-9

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Keywords

  • Polytope
  • Polytope diameter
  • Hirsch conjecture
  • Graph embedding
  • Prismatoid