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 , ∞ ) , ( 3 , 7 , 41 ) , ( 3 , 8 , 23 ) , ( 3 , 9 , 17 ) , ( 3 , 10 , 14 ) , ( 3 , 11 , 13 ) , ( 4 , 4 , ∞ ) , ( 4 , 5 , 19 ) , ( 4 , 6 , 11 ) , ( 4 , 7 , 9 ) , ( 5 , 5 , 9 ) , ( 5 , 6 , 7 ) . Jendrol’ (1999) improved this description, except for ( 4 , 4 , ∞ ) 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 , ∞ ) , ( 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 , ∞ ) , ( 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 , ∞ ) , ( 4 , 5 , 11 ) , ( 4 , 6 , 10 ) , ( 4 , 7 , 7 ) , ( 5 , 5 , 7 ) , ( 5 , 6 , 6 ) .