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The vertex-face weight of edges in 3-polytopes

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Abstract

The weight w(e) of an edge e in a 3-polytope is the maximum degree-sum of the two vertices and two faces incident with e. In 1940, Lebesgue proved that each 3-polytope without the so-called pyramidal edges has an edge e with w(e) ≤ 21. In 1995, this upper bound was improved to 20 by Avgustinovich and Borodin. Note that each edge of the n-pyramid is pyramidal and has weight n + 9. Recently, we constructed a 3-polytope without pyramidal edges satisfying w(e) ≥ 18 for each e. The purpose of this paper is to prove that each 3-polytope without pyramidal edges has an edge e with w(e) ≤ 18. In other terms, this means that each plane quadrangulation without a face incident with three vertices of degree 3 has a face with the vertex degree-sum at most 18, which is tight.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    O. V. Borodin

  2. Ammosov North-Eastern Federal University, Yakutsk, Russia

    A. O. Ivanova

Authors
  1. O. V. Borodin
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  2. A. O. Ivanova
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Correspondence to O. V. Borodin.

Additional information

The authors were supported by the Russian Foundation for Basic Research (Grants 12-01-00631 and 12-01-00448 for the first author and Grants 12-01-00631 and 12–01–98510 for the second author) 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.”

Novosibirsk and Yakutsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 2, pp. 338–350, March–April, 2015.

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Borodin, O.V., Ivanova, A.O. The vertex-face weight of edges in 3-polytopes. Sib Math J 56, 275–284 (2015). https://doi.org/10.1134/S003744661502007X

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  • DOI: https://doi.org/10.1134/S003744661502007X

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