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Sibirsk. Mat. Zh., 2017, Volume 58, Number 4, Pages 771–778 (Mi smj2896)  

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

Low and light $5$-stars in $3$-polytopes with minimum degree $5$ and restrictions on the degrees of major vertices

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

Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: In 1940, in attempts to solve the Four Color Problem, Henry Lebesgue gave an approximate description of the neighborhoods of $5$-vertices in the class $\mathbf P_5$ of 3-polytopes with minimum degree $5$. This description depends on $32$ main parameters. Very few precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in $\mathbf P_5$. Given a $3$-polytope $P$, denote the minimum of the maximum degrees (height) of the neighborhoods of $5$-vertices (minor $5$-stars) in $P$ by $h(P)$. Jendrol'and Madaras in 1996 showed that if a polytope $P$ in $\mathbf P_5$ is allowed to have a $5$-vertex adjacent to four $5$-vertices (called a minor $(5,5,5,5,\infty)$-star), then $h(P)$ can be arbitrarily large. For each $P^*$ in $\mathbf P_5$ with neither vertices of the degree from $6$ to $8$ nor minor $(5,5,5,5,\infty)$-star, it follows from Lebesgue's Theorem that $h(P^*)\le17$. We prove in particular that every such polytope $P^*$ satisfies $h(P^*)\le12$, and this bound is sharp. This result is best possible in the sense that if vertices of one of degrees in $\{6,7,8\}$ are allowed but those of the other two forbidden, then the height of minor $5$-stars in $\mathbf P_5$ under the absence of minor $(5,5,5,5,\infty)$-stars can reach $15$, $17$, or $14$, respectively.

Keywords: planar map, planar graph, $3$-polytope, structural properties, $5$-star, height, weight.

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.2017.58.405

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English version:
Siberian Mathematical Journal, 2017, 58:4, 600–605

Bibliographic databases:

Document Type: Article
UDC: 519.172.2
MSC: 35R30
Received: 20.10.2016

Citation: O. V. Borodin, A. O. Ivanova, D. V. Nikiforov, “Low and light $5$-stars in $3$-polytopes with minimum degree $5$ and restrictions on the degrees of major vertices”, Sibirsk. Mat. Zh., 58:4 (2017), 771–778; Siberian Math. J., 58:4 (2017), 600–605

Citation in format AMSBIB
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\paper Low and light $5$-stars in $3$-polytopes with minimum degree~$5$ and restrictions on the degrees of major vertices
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\pages 771--778
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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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