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Heights of Minor Faces in 3-Polytopes

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Abstract

Each 3-polytope has obviously a face \( f \) of degree \( d(f) \) at most 5 which is called minor. The height \( h(f) \) of \( f \) is the maximum degree of the vertices incident with \( f \). A type of a face \( f \) is defined by a set of upper constraints on the degrees of vertices incident with \( f \). This follows from the double \( n \)-pyramid and semiregular \( (3,3,3,n) \)-polytope, \( h(f) \) can be arbitrarily large for each \( f \) if a 3-polytope is allowed to have faces of types \( (4,4,\infty) \) or \( (3,3,3,\infty) \) which are called pyramidal. Denote the minimum height of minor faces in a given 3-polytope by \( h \). In 1996, Horňák and Jendrol’ proved that every 3-polytope without pyramidal faces satisfies \( h\leq 39 \) and constructed a 3-polytope with \( h=30 \). In 2018, we proved the sharp bound \( h\leq 30 \). In 1998, Borodin and Loparev proved that every 3-polytope with neither pyramidal faces nor \( (3,5,\infty) \)-faces has a face \( f \) such that \( h(f)\leq 20 \) if \( d(f)=3 \), or \( h(f)\leq 11 \) if \( d(f)=4 \), or \( h(f)\leq 5 \) if \( d(f)=5 \), where bounds 20 and 5 are best possible. We prove that every 3-polytope with neither pyramidal faces nor \( (3,5,\infty) \)-faces has \( f \) with \( h(f)\leq 20 \) if \( d(f)=3 \), or \( h(f)\leq 10 \) if \( d(f)=4 \), or \( h(f)\leq 5 \) if \( d(f)=5 \), where all bounds 20, 10, and 5 are best possible.

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Funding

This work was funded by the Russian Science Foundation (Grant 16–11–10054).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    O. V. Borodin & A. O. Ivanova

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

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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 2, pp. 250–268. https://doi.org/10.33048/smzh.2021.62.202

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Borodin, O.V., Ivanova, A.O. Heights of Minor Faces in 3-Polytopes. Sib Math J 62, 199–214 (2021). https://doi.org/10.1134/S0037446621020026

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