### Abstract

A basic combinatorial invariant of a convex polytope P is its f-vector f(P) = (f, f _{1} , ⋯ , 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.

Original language | English |
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Journal | Annals of Combinatorics |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

- Convex polytopes
- Face numbers

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

**The Numbers of Edges of 5-Polytopes with a Given Number of Vertices.** / Kusunoki, Takuya; Murai, Satoshi.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - The Numbers of Edges of 5-Polytopes with a Given Number of Vertices

AU - Kusunoki, Takuya

AU - Murai, Satoshi

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A basic combinatorial invariant of a convex polytope P is its f-vector f(P) = (f, f 1 , ⋯ , 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.

AB - A basic combinatorial invariant of a convex polytope P is its f-vector f(P) = (f, f 1 , ⋯ , 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.

KW - Convex polytopes

KW - Face numbers

UR - http://www.scopus.com/inward/record.url?scp=85061285289&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061285289&partnerID=8YFLogxK

U2 - 10.1007/s00026-019-00417-y

DO - 10.1007/s00026-019-00417-y

M3 - Article

JO - Annals of Combinatorics

JF - Annals of Combinatorics

SN - 0218-0006

ER -