Geodesic
On the Weight of Minor Faces in Triangle-Free 3-Polytopes
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 603-619

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The weight w(f) of a face f in a 3-polytope is the degree-sum of vertices incident with f. It follows from Lebesgue’s results of 1940 that every triangle-free 3-polytope without 4-faces incident with at least three 3-vertices has a 4-face with w ≤ 21 or a 5-face with w ≤ 17. 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.
Keywords: plane map, plane graph, 3-polytope, structural property, weight of face 
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Borodin, Oleg V.; Ivanova, Anna O. On the Weight of Minor Faces in Triangle-Free 3-Polytopes. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 603-619. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a6/