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 Sibirsk. Mat. Zh., 2017, Volume 58, Number 1, Pages 48–55 (Mi smj2838)

The height of faces of $3$-polytopes

O. V. Borodina, A. O. Ivanovab

a Sobolev Institute of Mathematics, Novosibirsk, Russia
b Ammosov North-Eastern Federal University, Yakutsk, Russia

Abstract: The height of a face in a $3$-polytope is the maximum degree of the incident vertices of the face, and the height of a $3$-polytope, $h$, is the minimum height of its faces. A face is pyramidal if it is either a $4$-face incident with three $3$-vertices, or a $3$-face incident with two vertices of degree at most $4$. If pyramidal faces are allowed, then $h$ can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated $3$-polytope has $h\le11$. In 1995, this bound was lowered by Avgustinovich and Borodin to $10$. Recently, we improved it to the sharp bound $8$. For plane triangulation without $4$-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that $h\le20$ which bound is sharp. Later, Borodin (1998) proved that $h\le20$ for all triangulated $3$-polytopes. Recently, we obtained the sharp bound $10$ for triangle-free $3$-polytopes. In 1996, Horňák and Jendrol' proved for arbitrarily $3$-polytopes that $h\le23$. In this paper we improve this bound to the sharp bound $20$.

Keywords: plane map, planar graph, $3$-polytope, structure properties, height of face.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-0586716-01-00499 Ministry of Education and Science of the Russian Federation ÍØ-1939.2014.1 The first author was supported by the Russian Foundation for Basic Research (Grants 15-01-05867 and 16-01-00499) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1939.2014.1). The second author worked within the governmental task “Organization of Scientific Research”.

DOI: https://doi.org/10.17377/smzh.2017.58.105

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English version:
Siberian Mathematical Journal, 2017, 58:1, 37–42

Bibliographic databases:

UDC: 519.17
MSC: 35R30

Citation: O. V. Borodin, A. O. Ivanova, “The height of faces of $3$-polytopes”, Sibirsk. Mat. Zh., 58:1 (2017), 48–55; Siberian Math. J., 58:1 (2017), 37–42

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Borodin O.V. Ivanova A.O., “New Results About the Structure of Plane Graphs: a Survey”, Proceedings of the 8th International Conference on Mathematical Modeling (ICMM-2017), AIP Conference Proceedings, 1907, ed. Egorov I. Popov S. Vabishchevich P. Antonov M. Lazarev N. Troeva M. Troeva M. Ivanova A. Grigorev Y., Amer Inst Physics, 2017, UNSP 030051
2. O. V. Borodin, M. A. Bykov, A. O. Ivanova, “More about the height of faces in 3-polytopes”, Discuss. Math. Graph Theory, 38:2 (2018), 443–453
3. O. V. Borodin, A. O. Ivanova, “Low faces of restricted degree in $3$-polytopes”, Siberian Math. J., 60:3 (2019), 405–411
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