Heights of Minor Faces in 3-Polytopes

Abstract

Each 3-polytope has obviously a face $ f $ of degree $ 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 on the 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 given 3-polytope by $ h $ . In 1996, Horňak and Jendrol’ proved that every 3-polytope without pyramidal faces satisfies $ h\leq 39 $ and constructed 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 with neither pyramidal faces nor $ (3,5,\infty) $ -faces has 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 all bounds 20, 10, and 5 are best possible.