The weight of faces in normal plane maps

Abstract

The weight of a face in a 3-polytope is the degree-sum of its incident vertices, and the weight of a 3-polytope, $w$, is the minimum weight 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 $w$ 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 $w\le 21$. 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 $w\le 29$, which bound is sharp. Later, Borodin (1998) proved that $w\le 29$ 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 $w\le 32$. In this paper we improve this bound to 30 and construct a polytope with $w=30$.