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Discussiones Mathematicae Graph Theory 16(1) (1996) 81-87
DOI: https://doi.org/10.7151/dmgt.1024

AN INEQUALITY CONCERNING EDGES OF MINOR WEIGHT IN CONVEX   3-POLYTOPES

Igor Fabrici Institute of Mathematics, Technical University Ilmenau PF 327, D-98684 Ilmenau, Germany e-mail: fabrici@mathematik.tu-ilmenau.de

Stanislav Jendrol' Department of Geometry and Algebra, P.J. afárik University Jesenná 5, 041 54 Koice, Slovak Republic e-mail: jendrol@Košice.upjs.sk

Dedicated to Professor E. Jucovic  on the occasion of his 70th birthday.

Abstract

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.

1991 Mathematics Subject Classification: 52B10, 52B05, 05C10.

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