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This article is cited in 5 scientific papers (total in 5 papers)
Discrete mathematics and mathematical cybernetics
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
Abstract:
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 $g\ge6$.
Keywords:
plane graph, structure properties, tight description, $3$-path, minimum degree, girth.
Received August 18, 2019, published September 27, 2019
Citation:
O. V. Borodin, A. O. Ivanova, “All tight descriptions of $3$-paths centered at $2$-vertices in plane graphs with girth at least $6$”, Sib. Èlektron. Mat. Izv., 16 (2019), 1334–1344
Linking options:
https://www.mathnet.ru/eng/semr1133 https://www.mathnet.ru/eng/semr/v16/p1334
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