Sib. Èlektron. Mat. Izv., 2018, Volume 15, Pages 1174–1181
Discrete mathematics and mathematical cybernetics
All tight descriptions of $3$-paths in plane graphs with girth at least $9$
V. A. Aksenova, O. V. Borodinb, A. O. Ivanovac
a Novosibirsk National Research University,
str. Pirogova, 1,
630090, Novosibirsk, Russia
b Sobolev Institute of Mathematics,
pr. Koptyuga, 4,
630090, Novosibirsk, Russia
c Ammosov North-Eastern Federal University,
str. Kulakovskogo, 48,
677000, Yakutsk, Russia
Lebesgue (1940) proved that every plane graph with minimum degree $\delta$ at least $3$ and girth $g$ 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.
Borodin and Ivanova (2015) gave seven tight descriptions of $3$-paths when $\delta\ge3$ and $g\ge4$. Furthermore, they 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, they 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, several 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 this paper, we prove ten new tight descriptions of $3$-paths for $\delta=2$ and $g\ge9$ and show that no other tight descriptions exist.
plane graph, structure properties, tight description, $3$-path, minimum degree, girth.
|Russian Foundation for Basic Research
|Ministry of Education and Science of the Russian Federation
|The first author was supported by the Russian Foundation for Basic Research (grant 18-01-00353). The second author was supported by the Russian Foundation for Basic Research (grant 16-01-00499). The third author’s work was performed as a part of government work “Leading researchers on an ongoing basis” (1.7217.2017/6.7).
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Received September 5, 2018, published October 16, 2018
V. A. Aksenov, O. V. Borodin, A. O. Ivanova, “All tight descriptions of $3$-paths in plane graphs with girth at least $9$”, Sib. Èlektron. Mat. Izv., 15 (2018), 1174–1181
Citation in format AMSBIB
\by V.~A.~Aksenov, O.~V.~Borodin, A.~O.~Ivanova
\paper All tight descriptions of $3$-paths in plane graphs with girth at least~$9$
\jour Sib. \`Elektron. Mat. Izv.
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