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Сиб. электрон. матем. изв., 2018, том 15, страницы 1174–1181
(Mi semr986)
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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
Дискретная математика и математическая кибернетика
All tight descriptions of $3$-paths in plane graphs with girth at least $9$
V. A. Aksenova, O. V. Borodinb, A. O. Ivanovac
a Novosibirsk National Research University,
str. Pirogova, 1,
630090, Novosibirsk, Russia
b Sobolev Institute of Mathematics,
pr. Koptyuga, 4,
630090, Novosibirsk, Russia
c Ammosov North-Eastern Federal University,
str. Kulakovskogo, 48,
677000, Yakutsk, Russia
Аннотация:
Lebesgue (1940) proved that every plane graph with minimum degree $\delta$ at least $3$ and girth $g$ 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.
Borodin and Ivanova (2015) gave seven tight descriptions of $3$-paths when $\delta\ge3$ and $g\ge4$. Furthermore, they 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, they 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, several 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 this paper, we prove ten new tight descriptions of $3$-paths for $\delta=2$ and $g\ge9$ and show that no other tight descriptions exist.
Ключевые слова:
plane graph, structure properties, tight description, $3$-path, minimum degree, girth.
| Финансовая поддержка |
Номер гранта |
Российский фонд фундаментальных исследований  |
18-01-00353_a
16-01-00499_a |
| Министерство образования и науки Российской Федерации |
1.7217.2017/6.7 |
| The first author was supported by the Russian Foundation for Basic Research (grant 18-01-00353). The second author was supported by the Russian Foundation for Basic Research (grant 16-01-00499). The third author’s work was performed as a part of government work “Leading researchers on an ongoing basis” (1.7217.2017/6.7). |
DOI:
https://doi.org/10.17377/semi.2018.15.095
 Полный текст:
PDF файл (147 kB)
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Тип публикации:
Статья
УДК:
519.172.2
MSC: 05C75
Поступила 5 сентября 2018 г., опубликована 16 октября 2018 г.
Язык публикации: английский
Образец цитирования:
V. A. Aksenov, O. V. Borodin, A. O. Ivanova, “All tight descriptions of $3$-paths in plane graphs with girth at least $9$”, Сиб. электрон. матем. изв., 15 (2018), 1174–1181
Цитирование в формате AMSBIB
\RBibitem{AksBorIva18}
\by V.~A.~Aksenov, O.~V.~Borodin, A.~O.~Ivanova
\paper All tight descriptions of $3$-paths in plane graphs with girth at least~$9$
\jour Сиб. электрон. матем. изв.
\yr 2018
\vol 15
\pages 1174--1181
\mathnet{http://mi.mathnet.ru/semr986}
\crossref{https://doi.org/10.17377/semi.2018.15.095}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000454860200037}
Образцы ссылок на эту страницу:
http://mi.mathnet.ru/semr986 http://mi.mathnet.ru/rus/semr/v15/p1174
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Эта публикация цитируется в следующих статьяx:
-
O. V. Borodin, A. O. Ivanova, “All tight descriptions of $3$-paths centered at $2$-vertices in plane graphs with girth at least $6$”, Сиб. электрон. матем. изв., 16 (2019), 1334–1344
 -
O. V. Borodin, A. O. Ivanova, “All tight descriptions of $3$-paths in plane graphs with girth at least $8$”, Сиб. электрон. матем. изв., 17 (2020), 496–501
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