Low edges in 3-polytopes

The height h(e)h(e) of an edge ee in a 3-polytope is the maximum degree of the two vertices and two faces incident with ee. In 1940, Lebesgue proved that every 3-polytope without so called pyramidal edges has an edge ee with h(e)≤11h(e)≤11. In 1995, this upper bound was improved to 10 by Avgustinovich and Borodin. Recently, we improved it to 9 and constructed a 3-polytope without pyramidal edges satisfying h(e)≥8h(e)≥8 for each ee.The purpose of this paper is to prove that every 3-polytope without pyramidal edges has an edge ee with h(e)≤8h(e)≤8.In different terms, this means that every plane quadrangulation without a face incident with three vertices of degree 3 has a face incident with a vertex of degree at most 8, which is tight.