All tight descriptions of 3-paths in plane graphs with girth at least 7

Abstract

Lebesgue (1940) proved that every plane graph with minimum degree δ 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 of 3-paths is tight if none of its parameter can be strengthened, and no triplet dropped.Borodin et al. (2013) gave a tight description of 3-paths in plane graphs with δ≥3 and g≥3, and another tight description was given by Borodin, Ivanova and Kostochka in 2017.In 2015, we gave seven tight descriptions of 3-paths when δ≥3 and g≥4. 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 δ≥3 and g≥3. The problem of producing all tight descriptions for g≥3 remains widely open even for δ≥3.Eleven tight descriptions of 3-paths were obtained for plane graphs with δ=2 and g≥4 by Jendrol’, Maceková, Montassier, and Soták, four of which are descriptions for g≥9. In 2018, Aksenov, Borodin and Ivanova proved nine new tight descriptions of 3-paths for δ=2 and g≥9 and showed that no other tight descriptions exist. Recently, we resolved the case g≥8.The purpose of this paper is to give a complete list of 15 tight descriptions of 3-paths in the plane graphs with δ=2 and g≥7.