An improvement of Lebesgue’s description of edges in 3-polytopes and faces in plane quadrangulations

Abstract

An edge e in a 3-polytope is of type (k1,k2,k3,k4) if the set of degrees of the vertices and faces incident with e is majorized by the vector (k1,k2,k3,k4).In 1940, Lebesgue proved that every 3-polytope has an edge of one of the types (3,3,3,∞), (3,3,4,11), (3,3,5,7), (3,4,4,5).This also provides a description of the faces of quadrangulated 3-polytopes in terms of degrees of their incident vertices.The purpose of our paper is to prove that every 3-polytope has an edge of one of the types (3,3,3,∞), (3,3,4,9), (3,3,5,6), (3,4,4,5), where all parameters except possibly 9 are best possible. We believe that 9 here is sharp and thus the whole description is tight.Our proof relies on the discharging method.