Describing faces in plane triangulations

Lebesgue (1940) proved that every plane triangulation contains a face with the vertex-degrees majorized by one of the following triples: (3, 6, infinity), (3, 7, 41), (3, 8, 23), (3, 9, 17), (3, 10, 14), (3, 11, 13), (4, 4, infinity), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), (5, 6, 7). Jendrol' (1999) improved this description, except for (4, 4, infinity) and (4, 6, 11), to (3, 4, 35), (3, 5, 21), (3, 6, 20), (3, 7, 16), (3, 8, 14), (3, 9, 14), (3, 10, 13), (4, 4, infinity), (4, 5, 13), (4, 6, 17), (4, 7, 8), (5, 5, 7), (5, 6, 6) and conjectured that the tight description is (3, 4, 30), (3, 5, 18), (3, 6, 20), (3, 7, 14), (3, 8, 14), (3, 9, 12), (3, 10,12), (4,4, infinity), (4, 5, 10), (4, 6, 15), (4, 7, 7), (5, 5, 7), (5, 6, 6). We prove that in fact every plane triangulation contains a face with the vertex-degrees majorized by one of the following triples, where every parameter is tight: (3, 4, 31), (3, 5, 21), (3, 6,20), (3, 7, 13), (3, 8, 14), (3, 9, 12), (3, 10, 12), (4,4, infinity), (4, 5, 11), (4, 6, 10), (4, 7, 7), (5, 5, 7), (5, 6, 6). (C) 2013 Elsevier B.V. All rights reserved.