- Tytuł:
- More About the Height of Faces in 3-Polytopes
- Autorzy:
-
Borodin, Oleg V.
Bykov, Mikhail A.
Ivanova, Anna O. - Powiązania:
- https://bibliotekanauki.pl/articles/31342325.pdf
- Data publikacji:
- 2018-05-01
- Wydawca:
- Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
- Tematy:
-
plane map
planar graph
3-polytope
structural properties
height of face - Opis:
- The height of a face in a 3-polytope is the maximum degree of its incident vertices, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large, so we assume the absence of pyramidal faces in what follows. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, Borodin and Ivanova improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20, which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily polytopes that h ≤ 23. Recently, Borodin and Ivanova obtained the sharp bounds 10 for trianglefree polytopes and 20 for arbitrary polytopes. In this paper we prove that any polytope has a face of degree at most 10 with height at most 20, where 10 and 20 are sharp.
- Źródło:
-
Discussiones Mathematicae Graph Theory; 2018, 38, 2; 443-453
2083-5892 - Pojawia się w:
- Discussiones Mathematicae Graph Theory
- Dostawca treści:
- Biblioteka Nauki
Artykuł