Discrete & Computational Geometry

, Volume 28, Issue 1, pp 29–44

Examples and Counterexamples for the Perles Conjecture

  • Haase
  • Ziegler
Article

DOI: 10.1007/s00454-001-0085-0

Cite this article as:
Haase & Ziegler Discrete Comput Geom (2002) 28: 29. doi:10.1007/s00454-001-0085-0

Abstract

The combinatorial structure of a d-dimensional simple convex polytope—as given, for example, by the set of the (d − 1)-regular subgraphs of the facets—can be reconstructed from its abstract graph. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists.

A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by Micha Perles: “The facet subgraphs of a simple d-polytope are exactly all the (d − 1)-regular, connected, induced, non-separating subgraphs.”

We present non-trivial classes of examples for the validity of the Perles conjecture: in particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes.

On the other hand, we observe that for any 4-dimensional counterexample, the boundary of the (simplicial) dual polytope PΔ contains a 2-complex without a free edge, and without 2-dimensional homology. Examples of such complexes are known; we use a modification of “Bing’s house” (two walls removed) to construct explicit 4-dimensional counterexamples to the Perles conjecture.

Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Haase
    • 1
  • Ziegler
    • 2
  1. 1.Department of Mathematics, Duke University, Durham, NC 27708-0320, USA haase@math.duke.eduUSA
  2. 2.MA 6-2, Institute of Mathematics, Technische Universität Berlin, D-10623 Berlin, Germany ziegler@math.tu-berlin.deGermany

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