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

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 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 delta >= 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 delta >= 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 delta >= 3 and g >= 3. The problem of producing all tight descriptions for g >= 3 remains widely open even for delta >= 3. Eleven tight descriptions of 3-paths were obtained for plane graphs with delta = 2 and g >= 4 by Jendrol', Macekova, Montassier, and Sotak, four of which are descriptions for g >= 9. In 2018, Aksenov, Borodin and Ivanova proved nine new tight descriptions of 3-paths for delta = 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 delta = 2 and g >= 7. (C) 2021 Elsevier B.V. All rights reserved.