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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2020, Volume 17, Pages 1516–1521
DOI: https://doi.org/10.33048/semi.2020.17.105
(Mi semr1299)
 

This article is cited in 3 scientific papers (total in 3 papers)

Discrete mathematics and mathematical cybernetics

An extension of Franklin's Theorem

O. V. Borodin, A. O. Ivanova

Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
Full-text PDF (333 kB) Citations (3)
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Abstract: Back in 1922, Franklin proved that every $3$-polytope with minimum degree $5$ has a $5$-vertex adjacent to two vertices of degree at most $6$, which is tight. This result has been extended and refined in several directions.
It is well-known that each $3$-polytope has a vertex of degree at most $5$, called minor vertex. A $3$-path $uvw$ is an $(i,j,k)$-path if $d(u)\le i$, $d(v)\le j$, and $d(w)\le k$, where $d(x)$ is the degree of a vertex $x$. A $3$-path is minor $3$-path if its central vertex is minor.
The purpose of this note is to extend Franklin' Theorem to the $3$-polytopes with minimum degree at least $4$ by proving that there exist precisely the following ten tight descriptions of minor $3$-paths:$\{(6,5,6),(4,4,9),(6,4,8),(7,4,7)\}$, $\{(6,5,6),(4,4,9),(7,4,8)\}$, $\{(6,5,6),(6,4,9),(7,4,7)\}$, $\{(6,5,6),(7,4,9)\}$, $\{(6,5,8),(4,4,9),(7,4,7)\}$,$\{(6,5,9),(7,4,7)\}$, $\{(7,5,7),(4,4,9),(6,4,8)\}$, $\{(7,5,7),(6,4,9)\}$,$\{(7,5,8),(4,4,9)\}$, and $\{(7,5,9)\}$.
Keywords: planar graph, plane map, $3$-polytope, structure properties, tight description, path, weight.
Funding agency Grant number
Russian Science Foundation 16-11-10054
This research was funded by the Russian Science Foundation (grant 16-11-10054).
Received March 27, 2020, published September 18, 2020
Bibliographic databases:
Document Type: Article
UDC: 519.172.2
MSC: 05C75
Language: English
Citation: O. V. Borodin, A. O. Ivanova, “An extension of Franklin's Theorem”, Sib. Èlektron. Mat. Izv., 17 (2020), 1516–1521
Citation in format AMSBIB
\Bibitem{BorIva20}
\by O.~V.~Borodin, A.~O.~Ivanova
\paper An extension of Franklin's Theorem
\jour Sib. \`Elektron. Mat. Izv.
\yr 2020
\vol 17
\pages 1516--1521
\mathnet{http://mi.mathnet.ru/semr1299}
\crossref{https://doi.org/10.33048/semi.2020.17.105}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000575248000001}
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  • https://www.mathnet.ru/eng/semr/v17/p1516
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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