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Sibirsk. Mat. Zh., 2018, Volume 59, Number 1, Pages 56–64 (Mi smj2953)  

This article is cited in 1 scientific paper (total in 1 paper)

Describing neighborhoods of $5$-vertices in a class of $3$-polytopes with minimum degree $5$

O. V. Borodin, A. O. Ivanova, D. V. Nikiforov

Sobolev Institute of Mathematics, Novosibirsk, Russia

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
$$ \begin{gathered} (6,6,7,7,7), (6,6,6,7,9), (6,6,6,6,11),
(5,6,7,7,8), (5,6,6,7,12), (5,6,6,8,10), (5,6,6,6,17),
(5,5,7,7,13), (5,5,7,8,10), (5,5,6,7,27), (5,5,6,6,\infty), (5,5,6,8,15), (5,5,6,9,11),
(5,5,5,7,41), (5,5,5,8,23), (5,5,5,9,17), (5,5,5,10,14), (5,5,5,11,13). \end{gathered} $$
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: $(5,6,6,5,\infty)$, $(5,6,6,6,15)$, and $(6,6,6,6,6)$, where all parameters are tight.

Keywords: plane graph, structure properties, $3$-polytope, neighborhood.

Funding Agency Grant Number
Russian Science Foundation 16-11-10054
The authors were funded by the Russian Science Foundation (Grant 16-11-10054).


DOI: https://doi.org/10.17377/smzh.2018.59.105

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English version:
Siberian Mathematical Journal, 2018, 59:1, 43–49

Bibliographic databases:

Document Type: Article
UDC: 519.17
MSC: 35R30
Received: 11.05.2017

Citation: O. V. Borodin, A. O. Ivanova, D. V. Nikiforov, “Describing neighborhoods of $5$-vertices in a class of $3$-polytopes with minimum degree $5$”, Sibirsk. Mat. Zh., 59:1 (2018), 56–64; Siberian Math. J., 59:1 (2018), 43–49

Citation in format AMSBIB
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\by O.~V.~Borodin, A.~O.~Ivanova, D.~V.~Nikiforov
\paper Describing neighborhoods of $5$-vertices in a~class of $3$-polytopes with minimum degree~$5$
\jour Sibirsk. Mat. Zh.
\yr 2018
\vol 59
\issue 1
\pages 56--64
\mathnet{http://mi.mathnet.ru/smj2953}
\crossref{https://doi.org/10.17377/smzh.2018.59.105}
\elib{http://elibrary.ru/item.asp?id=32824586}
\transl
\jour Siberian Math. J.
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\vol 59
\issue 1
\pages 43--49
\crossref{https://doi.org/10.1134/S0037446618010056}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. O. V. Borodin, A. O. Ivanova, “Light 3-stars in sparse plane graphs”, Sib. elektron. matem. izv., 15 (2018), 1344–1352  mathnet  crossref
  • Сибирский математический журнал Siberian Mathematical Journal
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