Describing Neighborhoods of 5-Vertices in a Class of 3-Polytopes with Minimum Degree 5
- Authors: Borodin O.V.1, Ivanova A.O.1, Nikiforov D.V.1
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Affiliations:
- Sobolev Institute of Mathematics
- Issue: Vol 59, No 1 (2018)
- Pages: 43-49
- Section: Article
- URL: https://journals.rcsi.science/0037-4466/article/view/171644
- DOI: https://doi.org/10.1134/S0037446618010056
- ID: 171644
Cite item
Abstract
Lebesgue proved in 1940 that each 3-polytope with minimum degree 5 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences
(6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11)
(5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17)
(5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6,∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11)
(5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13).
We prove that each 3-polytope with minimum degree 5 without vertices of degree from 7 to 10 contains a 5-vertex whose set of degrees of its neighbors is majorized by one of the following sequences: (5, 6, 6, 5, ∞), (5, 6, 6, 6, 15), and (6, 6, 6, 6, 6), where all parameters are tight.
Keywords
About the authors
O. V. Borodin
Sobolev Institute of Mathematics
Author for correspondence.
Email: brdnoleg@math.nsc.ru
Russian Federation, Novosibirsk
A. O. Ivanova
Sobolev Institute of Mathematics
Email: brdnoleg@math.nsc.ru
Russian Federation, Novosibirsk
D. V. Nikiforov
Sobolev Institute of Mathematics
Email: brdnoleg@math.nsc.ru
Russian Federation, Novosibirsk
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