The weight of faces in normal plane maps

The weight of a face in a 3-polytope is the degree-sum of its incident vertices, and the weight of a 3-polytope, ww, is the minimum weight of its faces. A face is pyramidal if it is either a 4-face incident with three 33-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then ww can be arbitrarily large, so we assume the absence of pyramidal faces in what follows.In 1940, Lebesgue proved that every quadrangulated 3-polytope has w≤21w≤21. In 1995, this bound was lowered by Avgustinovich and Borodin to 20. Recently, we improved it to the sharp bound 18.For plane triangulations without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that w≤29w≤29, which bound is sharp. Later, Borodin (1998) proved that w≤29w≤29 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 20 for triangle-free polytopes.In 1996, Horňák and Jendrol’ proved for arbitrarily polytopes that w≤32w≤32. In this paper we improve this bound to 30 and construct a polytope with w=30w=30.