All tight descriptions of 3-paths centered at 2-vertices in plane graphs with girth at least 6

摘要

Lebesgue (1940) proved that every plane graph with minimumdegree at least 3 and girth g (the length of a shortest cycle) at least 5has a path on three vertices (3-path) of degree 3 each. A description istight if no its parameter can be strengthened, and no triplet dropped.Borodin et al. (2013) gave a tight description of 3-paths in planegraphs with 3 and g 3, and another tight description was given byBorodin, Ivanova and Kostochka in 2017.In 2015, we gave seven tight descriptions of 3-paths when 3and g 4. Furthermore, we proved that this set of tight descriptionsis complete, which was a result of a new type in the structural theory ofplane graphs. Also, we characterized (2018) all one-term tight descriptionsif 3 and g 3. The problem of producing all tight descriptions forg 3 remains widely open even for 3.Recently, eleven tight descriptions of 3-paths were obtained for planegraphs with = 2 and g 4 by Jendrol’, Macekov′a, Montassier, andSot′ak, four of which descriptions are for g 9. In 2018, Aksenov, Borodinand Ivanova proved ten new tight descriptions of 3-paths for = 2 andg 9 and showed that no other tight descriptions exist.In this paper we give a complete list of tight descriptions of 3-pathscentered at a 2-vertex in the plane graphs with = 2 and g 6.