|
Сиб. электрон. матем. изв., 2020, том 17, страницы 496–501
(Mi semr1225)
|
|
|
|
Дискретная математика и математическая кибернетика
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
Аннотация:
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 $g\ge8$.
Ключевые слова:
Plane graph, structure properties, tight description, $3$-path, minimum degree, height, weight, girth.
Финансовая поддержка |
Номер гранта |
Российский фонд фундаментальных исследований  |
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). |
DOI:
https://doi.org/10.33048/semi.2020.17.030
Полный текст:
PDF файл (130 kB)
Список литературы:
PDF файл
HTML файл
Реферативные базы данных:
Тип публикации:
Статья
УДК:
519.172.2
MSC: 05C75 Поступила 4 марта 2020 г., опубликована 6 апреля 2020 г.
Язык публикации: английский
Образец цитирования:
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
Цитирование в формате AMSBIB
\RBibitem{BorIva20}
\by O.~V.~Borodin, A.~O.~Ivanova
\paper All tight descriptions of $3$-paths in plane graphs with girth at least~$8$
\jour Сиб. электрон. матем. изв.
\yr 2020
\vol 17
\pages 496--501
\mathnet{http://mi.mathnet.ru/semr1225}
\crossref{https://doi.org/10.33048/semi.2020.17.030}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000525532700001}
Образцы ссылок на эту страницу:
http://mi.mathnet.ru/semr1225 http://mi.mathnet.ru/rus/semr/v17/p496
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
|
Просмотров: |
Эта страница: | 62 | Полный текст: | 16 | Литература: | 2 |
|