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This article is cited in 8 scientific papers (total in 8 papers)
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.
Received: 24.11.2014
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
Linking options:
https://www.mathnet.ru/eng/smj2692 https://www.mathnet.ru/eng/smj/v56/i5/p982
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Abstract page: | 246 | Full-text PDF : | 60 | References: | 51 | First page: | 3 |
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