Аннотация:
Lebesgue (1940) proved that every plane
graph with minimum degree δ 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 δ⩾3 and g⩾3, and another tight
description was given by Borodin, Ivanova and Kostochka in 2017.
In 2015, we gave seven tight descriptions of 3-paths when
δ⩾3 and g⩾4. 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 δ⩾3 and
g⩾3. The problem of producing all tight descriptions for
g⩾3 remains widely open even for δ⩾3.
Recently, eleven tight descriptions of 3-paths were obtained for
plane graphs with δ=2 and g⩾4 by Jendrol',
Maceková, Montassier, and Soták, four of which
descriptions are for g⩾9. In 2018, Aksenov, Borodin and
Ivanova proved ten new tight descriptions of 3-paths for
δ=2 and g⩾9 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 δ=2
and g⩾6.
The first author was supported by the Russian Foundation
for Basic Research (grants 18-01-00353 and 16-01-00499) and
by Program of fundamental research of the SB RAS №
I.5.1, project No. 0314-2019-0016. The second author's work has
been supported by the Ministry of Science and Higher Education of
the Russian Federation (Grant No. FSRG-2017-0013).
Поступила18 августа 2019 г., опубликована 27 сентября 2019 г.
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
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\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}
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Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/semr1133
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Эта публикация цитируется в следующих 5 статьяx:
Andrew Pham, Bing Wei, “Independent bondage number of planar graphs with minimum degree at least 3”, Discrete Mathematics, 345:12 (2022), 113072
О. В. Бородин, А. О. Иванова, “Точное описание $3$-многогранников их старшими $3$-цепями”, Сиб. матем. журн., 62:3 (2021), 498–508; O. V. Borodin, A. O. Ivanova, “A tight description of $3$-polytopes by their major $3$-paths”, Siberian Math. J., 62:3 (2021), 400–408
O. V. Borodin, A. O. Ivanova, Ts. Ch.-D. Batueva, D. V. Nikiforov, “All Tight Descriptions of Major 3-Paths in 3-Polytopes Without 3-Vertices”, Sib. Electron. Math. Rep., 18 (2021), 456–463
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