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Сиб. электрон. матем. изв., 2019, том 16, страницы 1334–1344
(Mi semr1133)
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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 centered at $2$-vertices in plane graphs with girth at least $6$
O. V. Borodina, A. O. Ivanovab a Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, 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 ten new tight descriptions of $3$-paths for
$\delta=2$ and $g\ge9$ and showed that no other tight descriptions
exist.
In this paper we give a complete list of tight descriptions of
$3$-paths centered at a $2$-vertex in the plane graphs with $\delta=2$
and $g\ge6$.
Ключевые слова:
plane graph, structure properties, tight description, $3$-path, minimum degree, girth.
DOI:
https://doi.org/10.33048/semi.2019.16.092
Полный текст:
PDF файл (543 kB)
Список литературы:
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Реферативные базы данных:
Тип публикации:
Статья
УДК:
519.172.2
MSC: 05C75 Поступила 18 августа 2019 г., опубликована 27 сентября 2019 г.
Язык публикации: английский
Образец цитирования:
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
Цитирование в формате AMSBIB
\RBibitem{BorIva19}
\by O.~V.~Borodin, A.~O.~Ivanova
\paper All tight descriptions of $3$-paths centered at $2$-vertices in plane graphs with girth at least~$6$
\jour Сиб. электрон. матем. изв.
\yr 2019
\vol 16
\pages 1334--1344
\mathnet{http://mi.mathnet.ru/semr1133}
\crossref{https://doi.org/10.33048/semi.2019.16.092}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000488211800003}
Образцы ссылок на эту страницу:
http://mi.mathnet.ru/semr1133 http://mi.mathnet.ru/rus/semr/v16/p1334
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 in plane graphs with girth at least $8$”, Сиб. электрон. матем. изв., 17 (2020), 496–501
-
O. V. Borodin, A. O. Ivanova, “An extension of Franklin's Theorem”, Сиб. электрон. матем. изв., 17 (2020), 1516–1521
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