Elsevier

Discrete Mathematics

Volume 338, Issue 12, 6 December 2015, Pages 2234-2241
Discrete Mathematics

Low edges in 3-polytopes

https://doi.org/10.1016/j.disc.2015.05.018Get rights and content
Under an Elsevier user license
open archive

Abstract

The height h(e) of an edge e in a 3-polytope is the maximum degree of the two vertices and two faces incident with e. In 1940, Lebesgue proved that every 3-polytope without so called pyramidal edges has an edge e with h(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)8 for each e.

The purpose of this paper is to prove that every 3-polytope without pyramidal edges has an edge e with h(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.

Keywords

Plane map
Plane graph
3-polytope
Structural property
Height of edge

Cited by (0)