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) ≤ 11. 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) ≥ 8 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) ≤ 10 or a 5-face with h(f) ≤ 5, where the bounds 10 and 5 are sharp.
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Original Russian Text Copyright © 2015 Borodin O.V. and Ivanova A.O.
The first author was supported by the Russian Foundation for Basic Research (Grants 12-01-00631 and 15-01-05867) 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” and supported by the Russian Foundation for Basic Research (Grant 12-01-98510).
Novosibirsk; Yakutsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 5, pp. 982–988, September–October, 2015; DOI: 10.17377/smzh.2015.56.502
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Borodin, O.V., Ivanova, A.O. Heights of minor faces in triangle-free 3-polytopes. Sib Math J 56, 783–788 (2015). https://doi.org/10.1134/S003744661505002X
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DOI: https://doi.org/10.1134/S003744661505002X