Mathematics
Description of faces in 3-polytopes without vertices of degree from 4 to 9
A. O. Ivanova M. K. Ammosov North-Eastern Federal University, Kulakovskogo st., 48, Yakutsk 677000, Yakutia, Russia
Abstract:
In 1940, Lebesgue proved that every normal plane map contains a face for which the set of degrees of its vertices is majorized by one of the following sequences: (3, 6,
$\infty$), (3, 7, 41), (3, 8, 23), (3, 9, 17), (3, 10, 14), (3, 11, 13), (4, 4,
$\infty$), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), (5, 6, 7), (3, 3, 3,
$\infty$), (3, 3, 4, 11), (3, 3, 5, 7), (3, 4, 4, 5), (3, 3, 3, 3, 5). In this note prove that every 3-polytope without vertices of degree from 4 to 9 contains a face for which the set of degrees of its vertices is majorized by one of the following sequences: (3, 3,
$\infty$), (3, 10, 12), (3, 3, 3,
$\infty$), (3, 3, 3, 3, 3), which is tight.
Keywords:
planar graph, plane map, structure properties, 3-polytope, weight.
UDC:
519.172.2 Received: 07.05.2016