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ЖУРНАЛЫ // Сибирские электронные математические известия // Архив

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.

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



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