Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T11:21:31.228Z Has data issue: false hasContentIssue false
  • English
  • Français

The toroidal analogue to Eberhard's theorem

Published online by Cambridge University Press:  26 February 2010

Peter Gritzmann
Peter Gritzmann
Affiliation:
Math. Inst. Univ. Siegen, Hölderlinstrasse 3, D-5900 Siegen, Fed. Rep. Germany.
Get access

Extract

In 1891 Victor Eberhard proved the following theorem concerning the number pk(P) of k-gonal facets of simple polytopes P, [4].

Eberhard's Theorem. For each k ≥ 3, k ≠ 6, let pk be a non-negative integer. Then there exists a simple 3-polytope P such that pk(P) = pk (k ≠ 6), if, and only if,

Type
Research Article
Information
Mathematika , Volume 30 , Issue 2 , December 1983 , pp. 274 - 290
Copyright
Copyright © University College London 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Barnette, D.. Nonconvex vertices of toroidal polytopes. To appear.Google Scholar
2.Barnette, D., Jucovic, E. and Trenkler, M.. Toroidal maps with prescribed types of vertices and faces. Mathematika, 18 (1971), 82–90.CrossRefGoogle Scholar
3.Betke, U. and Gritzmann, P.. Polyedrische 2-Mannigfaltigkeiten mit wenigen nicht-konvexen Ecken. To appear in Monat. Math.Google Scholar
4.Eberhard, V.. Zur Morphologie der Polyeder (Teubner, Leipzig, 1891).Google Scholar
5.Gritzmann, P.. Polyedrische Realisierungen geschlossener 2-dimensionaler Mannigfaltigkeiten im ℝ3, Diss. (Siegen, 1980).Google Scholar
6.Gritzmann, P.. Upper and lower bounds of the valence-functional. Israel J. Math., 43 (1982), 237–243.CrossRefGoogle Scholar
7.Gritzmann, P.. Der Bereich der f-Vektoren für polyedrische Realisierungen des Torus. Mitt. math. Ges. Hamb., XI 2. To appear.Google Scholar
8.Gritzmann, P. and Höhne, R.. The non-existence of a number of classes of polyhedral 2-manifolds. To appear.Google Scholar
9.Grünbaum, B.. Convex polytopes (Interscience, New York, 1967).Google Scholar
10.Grünbaum, B.. Polytopes, graphs and complexes. Bull. Amer. Math. Soc., 76 (1970), 1131–1201.CrossRefGoogle Scholar
11.Grünbaum, B. and Shephard, G. C.. The theorems of Euler and Eberhard for tilings of the plane. Resultate Math., 5 (1982), 19–44.CrossRefGoogle Scholar
12.Jendrol, S. and Jucovic, E.. On the toroidal analogue of Eberhard's theorem. Proc. London Math. Soc. (3), 25 (1972), 385–398.CrossRefGoogle Scholar
13.Jendrol, S. and Jucovic, E.. On a conjecture by B. Grünbaum. Discr. Math., 2 (1972), 35–49.CrossRefGoogle Scholar
14.Jendrol, S. and Jucovic, E.. Generalization of a theorem by V. Eberhard. Math. Slovaca, 27 (1977), 383–407.Google Scholar
15.Jucovic, E. and Trenkler, M.. A theorem on the structure of cell-decompositions of orientable 2-manifolds. Mathematika, 20 (1973), 63–82.CrossRefGoogle Scholar
16.Malkevitch, J.. Properties of planar graphs with uniform vertex and face structure. Mem. Amer. Math. Soc., 99 (1970).Google Scholar
17.Simutis, J.. Geometric realisations of toroidal maps. Ph.D. thesis (University of California, Davis, 1977).Google Scholar
18.Zaks, J.. The analogue of Eberhard's theorem for 4-valent graphs on the torus. Israel J. Math., 9 (1971), 299–305.CrossRefGoogle Scholar