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The vertex-face weight of edges in 3-polytopes. (English. Russian original) Zbl 1331.52018
Sib. Math. J. 56, No. 2, 275-284 (2015); translation from Sib. Mat. Zh. 56, No. 2, 338-350 (2015).
Summary: The weight $$w(e)$$ of an edge $$e$$ in a 3-polytope is the maximum degree-sum of the two vertices and two faces incident with $$e$$. In 1940, Lebesgue proved that each 3-polytope without the so-called pyramidal edges has an edge $$e$$ with $$w(e) \leq 21$$. In 1995, this upper bound was improved to 20 by Avgustinovich and Borodin. Note that each edge of the $$n$$-pyramid is pyramidal and has weight $$n + 9$$. Recently, we constructed a 3-polytope without pyramidal edges satisfying $$w(e) \geq 18$$ for each $$e$$. The purpose of this paper is to prove that each 3-polytope without pyramidal edges has an edge $$e$$ with $$w(e) \leq 18$$. In other terms, this means that each plane quadrangulation without a face incident with three vertices of degree 3 has a face with the vertex degree-sum at most 18, which is tight.

##### MSC:
 52B10 Three-dimensional polytopes
Full Text:
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