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.
УДК:
519.172.2
MSC: 05C75 Поступила 4 марта 2020 г., опубликована
6 апреля 2020 г.
Язык публикации: английский
DOI:
10.33048/semi.2020.17.030