Geodesic
An inequality concerning edges of minor weight in convex 3-polytopes
Discussiones Mathematicae. Graph Theory, Tome 16 (1996) no. 1, pp. 81-87

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Let e_ij be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is 20e_3,3 + 25e_3,4 + 16e_3,5 + 10e_3,6 + 6[2/3]e_3,7 + 5e_3,8 + 2[1/2]e_3,9 + 2e_3,10 + 16[2/3]e_4,4 + 11e_4,5 + 5e_4,6 + 1[2/3]e_4,7 + 5[1/3]e_5,5 + 2e_5,6 ≥ 120; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.
Keywords: planar graph, convex 3-polytope, normal map 
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Fabrici, Igor; Jendrol', Stanislav. An inequality concerning edges of minor weight in convex 3-polytopes. Discussiones Mathematicae. Graph Theory, Tome 16 (1996) no. 1, pp. 81-87. http://geodesic.mathdoc.fr/item/DMGT_1996_16_1_a5/