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On the weight of minor faces in triangle-free 3-polytopes. (English) Zbl 1339.05112
Summary: The weight $$w(f)$$ of a face $$f$$ in a 3-polytope is the degree-sum of vertices incident with $$f$$. It follows from H. Lebesgue’s results [J. Math. Pures Appl., IX. Sér. 19, 27–43 (1940; Zbl 0024.28701)] that every triangle-free 3-polytope without 4-faces incident with at least three 3-vertices has a 4-face with $$w\leq21$$ or a 5-face with $$w\leq17$$. Here, the bound 17 is sharp, but it was still unknown whether 21 is sharp.
The purpose of this paper is to improve this 21 to 20, which is best possible.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C22 Signed and weighted graphs 52B99 Polytopes and polyhedra
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