RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Sibirsk. Mat. Zh.: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 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

Full text: PDF file (448 kB)
References: PDF file   HTML file

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
\Bibitem{BorIvaNik17} \by O.~V.~Borodin, A.~O.~Ivanova, D.~V.~Nikiforov \paper Low and light $5$-stars in $3$-polytopes with minimum degree~$5$ and restrictions on the degrees of major vertices \jour Sibirsk. Mat. Zh. \yr 2017 \vol 58 \issue 4 \pages 771--778 \mathnet{http://mi.mathnet.ru/smj2896} \crossref{https://doi.org/10.17377/smzh.2017.58.405} \elib{http://elibrary.ru/item.asp?id=29947448} \transl \jour Siberian Math. J. \yr 2017 \vol 58 \issue 4 \pages 600--605 \crossref{https://doi.org/10.1134/S003744661704005X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000408727100005} \elib{http://elibrary.ru/item.asp?id=31080377} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85028565991} 

Linking options:
• http://mi.mathnet.ru/eng/smj2896
• http://mi.mathnet.ru/eng/smj/v58/i4/p771

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 44 Full text: 4 References: 12 First page: 8

 Contact us: math-net2019_04 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2019