An inequality concerning edges of minor weight in convex 3-polytopes

• Volume: 16, Issue: 1, page 81-87
• ISSN: 2083-5892

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Abstract

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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 $20{e}_{3,3}+25{e}_{3,4}+16{e}_{3,5}+10{e}_{3,6}+6\left[2/3\right]{e}_{3,7}+5{e}_{3,8}+2\left[1/2\right]{e}_{3,9}+2{e}_{3,10}+16\left[2/3\right]{e}_{4,4}+11{e}_{4,5}+5{e}_{4,6}+1\left[2/3\right]{e}_{4,7}+5\left[1/3\right]{e}_{5,5}+2{e}_{5,6}\ge 120$; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

How to cite

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Igor Fabrici, and Stanislav Jendrol'. "An inequality concerning edges of minor weight in convex 3-polytopes." Discussiones Mathematicae Graph Theory 16.1 (1996): 81-87. <http://eudml.org/doc/270533>.

@article{IgorFabrici1996,
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.},
author = {Igor Fabrici, Stanislav Jendrol'},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {planar graph; convex 3-polytope; normal map; edge weights; planar maps; normal planar map; 3-connected planar map; Kotzig's theorem; Steinitz's theorem},
language = {eng},
number = {1},
pages = {81-87},
title = {An inequality concerning edges of minor weight in convex 3-polytopes},
url = {http://eudml.org/doc/270533},
volume = {16},
year = {1996},
}

TY - JOUR
AU - Igor Fabrici
AU - Stanislav Jendrol'
TI - An inequality concerning edges of minor weight in convex 3-polytopes
JO - Discussiones Mathematicae Graph Theory
PY - 1996
VL - 16
IS - 1
SP - 81
EP - 87
AB - 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.
LA - eng
KW - planar graph; convex 3-polytope; normal map; edge weights; planar maps; normal planar map; 3-connected planar map; Kotzig's theorem; Steinitz's theorem
UR - http://eudml.org/doc/270533
ER -

References

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