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

作者：Borodin Oleg V; Ivanova Anna O^*

来源：Discrete Mathematics, 2013, 313(23): 2841-2847.

DOI：10.1016/j.disc.2013.08.028

• Planar graph
• Plane map
• Structure properties
• Lebesgue%26apos
• s theorem
• Weight

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: %26lt;br%26gt;(4, 4, infinity), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), (5, 6, 7). %26lt;br%26gt;Borodin (2002) strengthened this to (4, 4, infinity), (4, 5, 17), (4, 6, 11), (4, 7, 8), (5, 5, 8), (5, 6, 6). %26lt;br%26gt;We obtain 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: %26lt;br%26gt;(4, 4, infinity), (4, 5, 14), (4, 6, 10), (4, 7, 7), (5, 5, 7), (5, 6, 6).

• 出版日期2013-12-6