All tight descriptions of $3$-paths in plane graphs with girth at least~$8$

O. V. Borodin ; A. O. Ivanova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 496-501

Résumé

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 $g\ge8$.

Détail

Keywords: Plane graph, structure properties, tight description, $3$-path, minimum degree, height, weight, girth.