Loading [MathJax]/jax/output/SVG/config.js
Sibirskii Matematicheskii Zhurnal
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

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


Sibirskii Matematicheskii Zhurnal, 2015, Volume 56, Number 5, Pages 982–987
DOI: https://doi.org/10.17377/smzh.2015.56.502
(Mi smj2692)
 

This article is cited in 8 scientific papers (total in 8 papers)

Heights of minor faces in triangle-free $3$-polytopes

O. V. Borodina, A. O. Ivanovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Ammosov North-Eastern Federal University, Yakutsk, Russia
Full-text PDF (491 kB) Citations (8)
References:
Abstract: The height $h(f)$ of a face $f$ in a $3$-polytope is the maximum of the degrees of vertices incident with $f$. A $4$-face is pyramidal if it is incident with at least three $3$-vertices. We note that in the $(3,3,3,n)$-Archimedean solid each face $f$ is pyramidal and satisfies $h(f)=n$.
In 1940, Lebesgue proved that every quadrangulated $3$-polytope without pyramidal faces has a face $f$ with $h(f)\le11$. In 1995, this bound was improved to $10$ by Avgustinovich and Borodin. Recently, the authors improved it to $8$ and constructed a quadrangulated $3$-polytope without pyramidal faces satisfying $h(f)\ge8$ for each $f$.
The purpose of this paper is to prove that each $3$-polytope without triangles and pyramidal $4$-faces has either a $4$-face with $h(f)\le10$ or a $5$-face with $h(f)\le5$, where the bounds $10$ and $5$ are sharp.
Keywords: plane map, plane graph, $3$-polytope, structural properties, height of a face.
Received: 24.11.2014
English version:
Siberian Mathematical Journal, 2015, Volume 56, Issue 5, Pages 783–788
DOI: https://doi.org/10.1134/S003744661505002X
Bibliographic databases:
Document Type: Article
UDC: 519.17
Language: Russian
Citation: O. V. Borodin, A. O. Ivanova, “Heights of minor faces in triangle-free $3$-polytopes”, Sibirsk. Mat. Zh., 56:5 (2015), 982–987; Siberian Math. J., 56:5 (2015), 783–788
Citation in format AMSBIB
\Bibitem{BorIva15}
\by O.~V.~Borodin, A.~O.~Ivanova
\paper Heights of minor faces in triangle-free $3$-polytopes
\jour Sibirsk. Mat. Zh.
\yr 2015
\vol 56
\issue 5
\pages 982--987
\mathnet{http://mi.mathnet.ru/smj2692}
\crossref{https://doi.org/10.17377/smzh.2015.56.502}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3492885}
\elib{https://elibrary.ru/item.asp?id=24817491}
\transl
\jour Siberian Math. J.
\yr 2015
\vol 56
\issue 5
\pages 783--788
\crossref{https://doi.org/10.1134/S003744661505002X}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000363722400002}
\elib{https://elibrary.ru/item.asp?id=24963224}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84944937623}
Linking options:
  • https://www.mathnet.ru/eng/smj2692
  • https://www.mathnet.ru/eng/smj/v56/i5/p982
  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Сибирский математический журнал Siberian Mathematical Journal
    Statistics & downloads:
    Abstract page:246
    Full-text PDF :60
    References:51
    First page:3
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2025