Israel Journal of Mathematics

, Volume 6, Issue 4, pp 398–411 | Cite as

Some analogues of eberhard’s theorem on convex polytopes

  • B. Grünbaum


Letp k denote the number ofk-gonal faces of a simple 3-polytope. Euler’s relation leads to an equation between thep k ’s which does not involvep 6. Eberhard proved in 1891 that every sequence of non-negative integers (p 3,p 4,…) satisfying this equation corresponds to a polytope for suitable values ofp 6. In the present paper it is established that ifp 3=p 4=0 then every valuep 6≧8 is suitable.


Planar Graph Nonnegative Integer Convex POLYTOPES Local Insertion Topological Disc 
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  1. 1.
    D. W. Barnette,On p-vectors of simple 3-polytopes. J. Combinatorial Theory (to appear).Google Scholar
  2. 2.
    D. W. Barnette and B. Grünbaum,On Steinitz’s theorem concerning convex 3-polytopes and on some properties of 3-connected graphs. To appear in “The many facets of graph theory”, edited by G. Chartrand and S. F. Kapoor, Springer, 1969.Google Scholar
  3. 3.
    M. Brückner,Vielecke und Vielflache. Leipzig 1900.Google Scholar
  4. 4.
    V. Eberhard,Zur Morphologie der Polyeder. Leipzig 1891.Google Scholar
  5. 5.
    B. Grünbaum,Convex polytopes. New York 1967.Google Scholar
  6. 6.
    B. Grünbaum and T. S. Motzkin,The number of hexagons and the simplicity of geodesics on certain polyhedra. Canad. J. Math.,15 (1963) 744–751.zbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Malkevitch,Properties of planar graphs with uniform vertex and face structure. Ph.D. Thesis, University of Wisconsin 1968.Google Scholar
  8. 8.
    E Steinitz,Polyeder und Raumeinteilungen. Enzykl. math. Wiss., Vol. 3 (Geometrie) Part 3AB12, pp 1–139 (1922).Google Scholar
  9. 9.
    E. Steinitz and H. Rademacher,Vorlesungen über die Theorie der Polyeder. Berlin 1934.Google Scholar

Copyright information

© Hebrew University 1968

Authors and Affiliations

  • B. Grünbaum
    • 1
  1. 1.University of WashingtonSeattle

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