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. 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.

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