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 Sibirsk. Mat. Zh., 2015, Volume 56, Number 5, Pages 982–987 (Mi smj2692)

Heights of minor faces in triangle-free $3$-polytopes

O. V. Borodina, A. O. Ivanovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Ammosov North-Eastern Federal University, Yakutsk, Russia

Abstract: The height $h(f)$ of a face $f$ in a $3$-polytope is the maximum of the degrees of vertices incident with $f$. A $4$-face is pyramidal if it is incident with at least three $3$-vertices. We note that in the $(3,3,3,n)$-Archimedean solid each face $f$ is pyramidal and satisfies $h(f)=n$.
In 1940, Lebesgue proved that every quadrangulated $3$-polytope without pyramidal faces has a face $f$ with $h(f)\le11$. In 1995, this bound was improved to $10$ by Avgustinovich and Borodin. Recently, the authors improved it to $8$ and constructed a quadrangulated $3$-polytope without pyramidal faces satisfying $h(f)\ge8$ for each $f$.
The purpose of this paper is to prove that each $3$-polytope without triangles and pyramidal $4$-faces has either a $4$-face with $h(f)\le10$ or a $5$-face with $h(f)\le5$, where the bounds $10$ and $5$ are sharp.

Keywords: plane map, plane graph, $3$-polytope, structural properties, height of a face.

DOI: https://doi.org/10.17377/smzh.2015.56.502

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English version:
Siberian Mathematical Journal, 2015, 56:5, 783–788

Bibliographic databases:

UDC: 519.17

Citation: O. V. Borodin, A. O. Ivanova, “Heights of minor faces in triangle-free $3$-polytopes”, Sibirsk. Mat. Zh., 56:5 (2015), 982–987; Siberian Math. J., 56:5 (2015), 783–788

Citation in format AMSBIB
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This publication is cited in the following articles:
1. O. V. Borodin, A. O. Ivanova, E. I. Vasil'eva, “A Steinberg-like approach to describing faces in $3$-polytopes”, Graphs Comb., 33:1 (2017), 63–71
2. O. V. Borodin, A. O. Ivanova, “New results about the structure of plane graphs: a survey”, Proceedings of the 8th International Conference on Mathematical Modeling, ICMM-2017, AIP Conf. Proc., 1907, eds. I. Egorov, S. Popov, P. Vabishchevich, M. Antonov, N. Lazarev, M. Troeva, M. Troeva, A. Ivanova, Y. , Amer. Inst. Phys., 2017, UNSP 030051
3. 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
4. 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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