Geodesic
The weight of edge in 3-polytopes
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 11 (2014), pp. 457-463 Cet article a éte moissonné depuis la source Math-Net.Ru

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The height of an edge in 3-polytopes is the maximum degree of its incident vertices and faces. In 1940, Lebesgue proved that each 3-polytope without pyramidal edges has an edge of height at most 11. This upper bound was lowered to 10 by Avgustinovich and Borodin (1995). The best known lower bound for the height of edges is 7. We lower upper bound to 9 and give a construction of 3-polytope which has no edges of height smaller than 8.
Keywords: planar map, planar graph, 3-polytope, structural properties, height. 
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O. V. Borodin; A. O. Ivanova. The weight of edge in 3-polytopes. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 11 (2014), pp. 457-463. http://geodesic.mathdoc.fr/item/SEMR_2014_11_a51/

[1] S. V. Avgustinovich, O. V. Borodin, “Okrestnosti reber v normalnykh kartakh”, Diskretnyi analiz i issledovanie operatsii, 2:2–3 (1995), 3–9 | MR

[2] H. Lebesgue, “Quelques conséquences simples de la formule d'Euler”, J. Math. Pures Appl., 19 (1940), 27–43 | MR

[3] E. Steinitz, “Polyheder und Raumeinteilungen”, Enzykl. math. Wiss. (Geometrie), 3AB, no. 12, 1922, 1–139