Describing 3-faces in normal plane maps with minimum degree 4

In 1940, Lebesgue proved that every 3-polytope with minimum degree at least 4 contains a 3-face for which the set of degrees of its vertices is majorized by one of the following sequences: ( 4 , 4 , ∞ ) , ( 4 , 5 , 19 ) , ( 4 , 6 , 11 ) , ( 4 , 7 , 9 ) , ( 5 , 5 , 9 ) , ( 5 , 6 , 7 ) . n (2002) strengthened this to ( 4 , 4 , ∞ ) , ( 4 , 5 , 17 ) , ( 4 , 6 , 11 ) , ( 4 , 7 , 8 ) , ( 5 , 5 , 8 ) , ( 5 , 6 , 6 ) . ain the following description of 3-faces in normal plane maps with minimum degree at least 4 (in particular, it holds for 3-polytopes) in which every parameter is best possible and is attained independently of the others: ( 4 , 4 , ∞ ) , ( 4 , 5 , 14 ) , ( 4 , 6 , 10 ) , ( 4 , 7 , 7 ) , ( 5 , 5 , 7 ) , ( 5 , 6 , 6 ) .