Combinatorial structure of faces in triangulated 3-polytopes with minimum degree 4

Abstract

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 entries: (4, 4,∞), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), and (5, 6, 7). This description was strengthened by Borodin (2002) to (4, 4,∞), (4, 5, 17), (4, 6, 11), (4, 7, 8), (5, 5, 8), and (5, 6, 6). For triangulations with minimum degree at least 4, Jendrol’ (1999) gave a description of faces: (4, 4,∞), (4, 5, 13), (4, 6, 17), (4, 7, 8), (5, 5, 7), and (5, 6, 6). We obtain the following description of faces in plane triangulations (in particular, for triangulated 3-polytopes) with minimum degree at least 4 in which all parameters are best possible and are attained independently of the others: (4, 4,∞), (4, 5, 11), (4, 6, 10), (4, 7, 7), (5, 5, 7), and (5, 6, 6). In particular, we disprove a conjecture by Jendrol’ (1999) on the combinatorial structure of faces in triangulated 3-polytopes.