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

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Abstract

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.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka , 565-0871, Japan

    Takuya Kusunoki

  2. Department of Mathematics, Faculty of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku, Tokyo, 169-8050, Japan

    Satoshi Murai

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  1. Takuya Kusunoki
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  2. Satoshi Murai
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Correspondence to Satoshi Murai.

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Kusunoki, T., Murai, S. The Numbers of Edges of 5-Polytopes with a Given Number of Vertices. Ann. Comb. 23, 89–101 (2019). https://doi.org/10.1007/s00026-019-00417-y

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  • DOI: https://doi.org/10.1007/s00026-019-00417-y

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