Skip to main content
SpringerLink
Log in

Long monotone paths on simple 4-polytopes

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

TheMonotone Upper Bound Problem (Klee, 1965) asks if the maximal numberM(d,n) of vertices in a monotone path along edges of ad-dimensional polytope withn facets can be as large as conceivably possible: IsM(d,n)=M ubt (d,n), the maximal number of vertices that ad-polytope withn facets can have according to the Upper Bound Theorem?

We show that in dimensiond=4, the answer is “yes”, despite the fact that it is “no” if we restrict ourselves to the dual-to-cyclic polytopes. For eachn≥5, we exhibit a realization of a polar-to-neighborly 4-dimensional polytope withn facets and a Hamilton path through its vertices that is monotone with respect to a linear objective function.

This constrasts an earlier result, by which no polar-to-neighborly 6-dimensional polytope with 9 facets admits a monotone Hamilton path.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. W. Barnette,A family of neighborly polytopes, Israel Journal of Mathematics39 (1981), 127–140.

    MATH  MathSciNet  Google Scholar 

  2. F. Holt and V. Klee,A proof of the strict monotone 4-step conjecture, Contemporary Mathematics223 (1999), 201–216.

    MathSciNet  Google Scholar 

  3. M. Joswig, V. Kaibel and F. Körner,On the k-systems of a simple polytope, Israel Journal of Mathematics129 (2002), 109–117.

    MATH  MathSciNet  Google Scholar 

  4. V. Klee,Heights of convex polytopes, Journal of Mathematical Analysis and Applications11 (1965), 176–190.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. McMullen,The numbers of faces of simplicial polytopes, Israel Journal of Mathematics9 (1971), 559–570.

    MATH  MathSciNet  Google Scholar 

  6. T. S. Motzkin,Comonotone curves and polyhedra, Abstract, Bulletin of the American Mathematical Society63 (1957), 35.

    Google Scholar 

  7. J. Pfeifle and G. M. Ziegler,On the monotone upper bound problem, Experimental Mathematics13 (2004) 1–11.

    MATH  MathSciNet  Google Scholar 

  8. I. Shemer,Neighborly polytopes, Israel Journal of Mathematics43 (1982), 291–314.

    MATH  MathSciNet  Google Scholar 

  9. G. M. Ziegler,Lectures on Polytopes, Graduate Texts in Mathematics152, Springer, New York, 1995. Revised edition 1998.

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Julian Pfeifle

    Present address: Department de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Edifici Omega, Campus Nord, Jordi Girona 1-3, E-08034, Barcelona, Spain

Authors and Affiliations

  1. Institut de Matemàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, E-08007, Barcelona, Spain

    Julian Pfeifle

Authors
  1. Julian Pfeifle
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The author was financed in part by the DFG GraduiertenkollegCombinatorics, Geometry, and Computation (GRK 588-2), the GIF projectCombinatorics of Polytopes in Euclidean Spaces (I-624-35.6/1999), and post-doctoral fellowships from MSRI and Institut de Matemàtica de la Universitat de Barcelona.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pfeifle, J. Long monotone paths on simple 4-polytopes. Isr. J. Math. 150, 333–355 (2005). https://doi.org/10.1007/BF02762386

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02762386

Keywords

Search

Navigation