Low edges in 3-polytopes

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...

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Bibliographic Details
Published inDiscrete mathematics Vol. 338; no. 12; pp. 2234 - 2241
Main Authors Borodin, O.V., Ivanova, A.O.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 06.12.2015
Subjects
3-polytope
Construction
Graph theory
Height of edge
Mathematical analysis
Plane graph
Plane map
Planes
Structural property
Upper bounds
Height of edge
Plane map
Structural property
3-polytope
Plane graph
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 23
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2015.05.018