Реферат: Each 3-polytope has obviously a face $ f $ ofdegree $ 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 onthe 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 given3-polytope by $ h $. In 1996, Horňák and Jendrol’ proved that every3-polytope without pyramidal faces satisfies $ h\leq 39 $ andconstructed 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 withneither pyramidal faces nor $ (3,5,\infty) $-faceshas 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 allbounds 20, 10, and 5 are best possible. © 2021, Pleiades Publishing, Ltd.

Библиографическая ссылка: Borodin O.V. , Ivanova A.O.
Heights of Minor Faces in 3-Polytopes
Siberian Mathematical Journal. 2021. V.62. N2. P.199-214. DOI: 10.1134/S0037446621020026 WOS Scopus OpenAlex

Идентификаторы БД:

Web of science:	WOS:000635714500002
Scopus:	2-s2.0-85103969954
OpenAlex:	W3148816826

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