Abstract
A polytope P with 2n vertices is called equipartite if for any partition of its vertex set into two equal-size sets V 1 and V 2, there is an isometry of the polytope P that maps V 1 onto V 2. We prove that an equipartite polytope in ℝd can have at most 2d+2 vertices. We show that this bound is sharp and identify all known equipartite polytopes in ℝd. We conjecture that the list is complete.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. J. Cameron, P.M. Neumann and J. Saxl, An interchange property in finite permutation groups, The Bulletin of the London Mathematical Society 11 (1979), 161–169.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, Springer-Verlag, Berlin-New York, 1980.
H. M. Cundy and A. P. Rollett, Mathematical Models, 2nd edition, Clarendon Press, Oxford, 1961.
B. Grünbaum, Convex Polytopes, Second Edition, Springer, Berlin, 2003.
B. Grünbaum, T. Kaiser, D. Král’ and M. Rosenfeld, Equipartite graphs, Israel Journal of Mathematics 168 (2008), 431–444.
B. Grünbaum and G. C. Shephard, Spherical tilings with transitivity properties, in The Geometric Vein — The Coxeter Festschrift (C. Davis et al., eds.), Springer-Verlag, New York, 1981, pp. 65–98.
N. Jacobson, Basic algebra I, 2nd edition, W. H. Freeman, New York, 1985.
M. Molloy and B. Reed, Graph Colouring and the Probabilistic Method, Algorithms and Combinatorics, Vol. 23, Springer-Verlag, Berlin, 2002.
S. A. Robertson, Polytopes and Symmetry, Journal of the London Mathematical Society Lecture Note Series #90, Cambridge University Press, Cambridge, 1984.
D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
Author information
Authors and Affiliations
Corresponding author
Additional information
Support from the grant Kontakt ME 885 of the Czech Ministry of Education is gratefully acknowledged.
Supported by Research Plan MSM 4977751301 of the Czech Ministry of Education.
The institute is supported by Ministry of Education of Czech Republic as project 1M0545.
The author would like to acknowledge support by the Fulbright Senior Specialist Program.
Rights and permissions
About this article
Cite this article
Grünbaum, B., Kaiser, T., Král’, D. et al. Equipartite polytopes. Isr. J. Math. 179, 235–252 (2010). https://doi.org/10.1007/s11856-010-0080-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-010-0080-3