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

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',

Macekov\'{a}, Montassier, and Sot\'{a}k, 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 $g\ge8$.