The height of faces of 3-polytopes

Сибирский математический журнал

A. O. Ivanova¹  O. V. Borodin² 

[1] Ammosov North-Eastern Federal University;Sobolev Institute of Mathematics

关键词: plane map, planar graph, 3-polytope, structure properties, height of face ;   

DOI  :  10.1134/S0037446617010050

学科分类：数学（综合）

来源: Izdatel stvo Instituta Matematiki Rossiiskoi Akademii Nauk

【 摘 要 】

The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20 which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 10 for triangle-free 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily 3-polytopes that h ≤ 23. In this paper we improve this bound to the sharp bound 20.

【 授权许可】

CC BY