Skip to main content
SpringerLink
Log in

Simple polytopes without small separators, II: Thurston’s bound

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

Abstract

We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least Ω(n/log n). This establishes a strong form of a claim by Thurston, for which the construction and proof had been lost.

We construct the polytopes by cutting off the vertices and then the edges of a particular type of neighborly cubical polytopes. The graphs of simple polytopes thus obtained are 4-regular; they contain 3-regular “cube-connected cycle graphs” as minors of spanning subgraphs.

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. M. Joswig and T. Rörig, Neighborly cubical polytopes and spheres, Israel Journal of Mathematics 159 (2007), 221–242.

    Article  MathSciNet  Google Scholar 

  2. M. Joswig and G. M. Ziegler, Neighborly cubical polytopes, Discrete & Computational Geometry 24 (2000), 325–344.

    Article  MathSciNet  Google Scholar 

  3. V. Kaibel, On the expansion of graphs of 0/1-polytopes, in The Sharpest Cut, MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA, 2004, pp. 199–216.

    Chapter  Google Scholar 

  4. G. Kalai, Polytope skeletons and paths, in Handbook of Discrete and Computational Geometry, CRC Press Series on Discrete Mathematics and its Applications, CRC, Boca Raton, FL, First edition 1997, pp. 331–344.

    MATH  Google Scholar 

  5. R. J. Lipton and R. E. Tarjan, A separator theorem for planar graphs, SIAM Journal on Applied Mathematics 36 (1979), 177–189.

    Article  MathSciNet  Google Scholar 

  6. L. Loiskekoski and G. M. Ziegler, Simple polytopes without small separators, Israel Journal of Mathematics 221 (2017), 731–739.

    Article  MathSciNet  Google Scholar 

  7. G. L. Miller, S.-H. Teng, W. Thurston and S. A. Vavasis, Separators for sphere-packings and nearest neighbor graphs, Journal of the ACM 44 (1997), 1–29.

    Article  MathSciNet  Google Scholar 

  8. F. P. Preparata and J. Vuillemin, The cube-connected cycles: a versatile network for parallel computation, Communications of the ACM 24 (1981), 300–309.

    Article  MathSciNet  Google Scholar 

  9. G. Ringel, Über drei kombinatorische Probleme am n-dimensionalen Würfel und Würfelgitter, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 20 (1955), 10–19.

    Article  MathSciNet  Google Scholar 

  10. R. Sanyal and G. M. Ziegler, Construction and analysis of projected deformed products, Discrete & Computational Geometry 43 (2010), 412–435.

    Article  MathSciNet  Google Scholar 

  11. A. Sinclair, Algorithms for Random Generation and Counting: A Markov Chain Approach, Progress in Theoretical Computer Science, Birkhäuser Boston, Boston, MA, 1993.

    Google Scholar 

  12. G. M. Ziegler, Polyhedral surfaces of high genus, in Discrete Differential Geometry, Oberwolfach Seminars, Vol. 38, Birkhäuser, Basel, 2008, pp. 191–213.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, FU Berlin, Arnimallee 2, 14195, Berlin, Germany

    Lauri Loiskekoski & Günter M. Ziegler

Authors
  1. Lauri Loiskekoski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Günter M. Ziegler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Günter M. Ziegler.

Additional information

The first author was funded by DFG through the Berlin Mathematical School. Research by the second author was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Loiskekoski, L., Ziegler, G.M. Simple polytopes without small separators, II: Thurston’s bound. Isr. J. Math. 228, 293–303 (2018). https://doi.org/10.1007/s11856-018-1764-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1764-3

Search

Navigation