The Numbers of Edges of 5-Polytopes with a Given Number of Vertices

  • Takuya Kusunoki
  • Satoshi MuraiEmail author


A basic combinatorial invariant of a convex polytope P is its f-vector \(f(P)=(f_0,f_1,\dots ,f_{\dim P-1})\), where \(f_i\) is the number of i-dimensional faces of P. Steinitz characterized all possible f-vectors of 3-polytopes and Grünbaum characterized the pairs given by the first two entries of the f-vectors of 4-polytopes. In this paper, we characterize the pairs given by the first two entries of the f-vectors of 5-polytopes. The same result was also proved by Pineda-Villavicencio, Ugon and Yost independently.


Convex polytopes Face numbers 

Mathematics Subject Classification

52B05 52B11 



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Authors and Affiliations

  1. 1.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversitySuitaJapan
  2. 2.Department of Mathematics, Faculty of EducationWaseda UniversityTokyoJapan

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