Low Faces of Restricted Degree in 3-Polytopes

Abstract

The degree of a vertex or face in a 3-polytope is the number of incident edges. A k-face is one of degree k, a k−-face has degree at most k. The height of a face is the maximum degree of its incident vertices; 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; and so we assume the absence of pyramidal faces in what follows. In 1940, Lebesgue proved that each quadrangulated 3-polytope has a face f with h(f) ≤ 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 proved that h ≤ 20 for all triangulated 3-polytopes. In 1996, Horňak and Jendrol’ proved for arbitrarily polytopes that h ≤ 23. Recently, we obtained the sharp bounds h ≤ 10 for triangle-free polytopes and h ≤ 20 for arbitrary polytopes. Later, Borodin, Bykov, and Ivanova refined the latter result by proving that any polytope has a 10−-face of height at most 20, where 10 and 20 are sharp. Also, we proved that any polytope has a 5−-face of height at most 30, where 30 is sharp and improves the upper bound of 39 obtained by Horňak and Jendrol’ (1996). In this paper we prove that every polytope has a 6−-face of height at most 22, where 6 and 22 are best possible. Since there is a construction in which every face of degree from 6 to 9 has height 22, we now know everything concerning the maximum heights of restricted-degree faces in 3-polytopes.