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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2020, Volume 17, Pages 496–501
DOI: https://doi.org/10.33048/semi.2020.17.030
(Mi semr1225)
 

Discrete mathematics and mathematical cybernetics

All tight descriptions of $3$-paths in plane graphs with girth at least $8$

O. V. Borodina, A. O. Ivanovab

a Sobolev Institute of Mathematics, 4, Koptyuga ave., 630090, Novosibirsk, Russia
b Ammosov North-Eastern Federal University, 48, Kulakovskogo str., Yakutsk, 677000, Russia
References:
Abstract: Lebesgue (1940) proved that every plane graph with minimum degree $\delta$ at least 3 and girth $g$ (the length of a shortest cycle) at least $5$ has a path on three vertices ($3$-path) of degree $3$ each. A description is tight if no its parameter can be strengthened, and no triplet dropped.
Borodin et al. (2013) gave a tight description of $3$-paths in plane graphs with $\delta\ge3$ and $g\ge3$, and another tight description was given by Borodin, Ivanova and Kostochka in 2017.
In 2015, we gave seven tight descriptions of $3$-paths when $\delta\ge3$ and $g\ge4$. Furthermore, we proved that this set of tight descriptions is complete, which was a result of a new type in the structural theory of plane graphs. Also, we characterized (2018) all one-term tight descriptions if $\delta\ge3$ and $g\ge3$. The problem of producing all tight descriptions for $g\ge3$ remains widely open even for $\delta\ge3$.
Recently, eleven tight descriptions of $3$-paths were obtained for plane graphs with $\delta=2$ and $g\ge4$ by Jendrol', Maceková, Montassier, and Soták, four of which descriptions are for $g\ge9$. In 2018, Aksenov, Borodin and Ivanova proved nine new tight descriptions of $3$-paths for $\delta=2$ and $g\ge9$ and showed that no other tight descriptions exist.
The purpose of this note is to give a complete list of tight descriptions of $3$-paths in the plane graphs with $\delta=2$ and $g\ge8$.
Keywords: Plane graph, structure properties, tight description, $3$-path, minimum degree, height, weight, girth.
Funding agency Grant number
Russian Foundation for Basic Research 18-01-00353_a
19-01-00682
The first author' work was supported by Mathematical Center in Akademgorodok. The second author' work was supported the Russian Foundation for Basic Research (grants 18-01-00353 and 19-01-00682).
Received March 4, 2020, published April 6, 2020
Bibliographic databases:
Document Type: Article
UDC: 519.172.2
MSC: 05C75
Language: English
Citation: O. V. Borodin, A. O. Ivanova, “All tight descriptions of $3$-paths in plane graphs with girth at least $8$”, Sib. Èlektron. Mat. Izv., 17 (2020), 496–501
Citation in format AMSBIB
\Bibitem{BorIva20}
\by O.~V.~Borodin, A.~O.~Ivanova
\paper All tight descriptions of $3$-paths in plane graphs with girth at least~$8$
\jour Sib. \`Elektron. Mat. Izv.
\yr 2020
\vol 17
\pages 496--501
\mathnet{http://mi.mathnet.ru/semr1225}
\crossref{https://doi.org/10.33048/semi.2020.17.030}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000525532700001}
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