# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## 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 eij 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 20e3,3+25e3,4+16e3,5+10e3,6+6[2/3] e3,7+5e3,8+2[1/2] e3,9+2e3,10+16[2/3] e4,4+11e4,5+5e4,6+1[2/3]e4,7+5[1/3] e5,5+2e5,6 ≥ 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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