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Эта публикация цитируется в 7 научных статьях (всего в 7 статьях)
Дискретная математика и математическая кибернетика
Light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5
O. V. Borodina, A. O. Ivanovab a Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
b Ammosov North-Eastern Federal University, str. Kulakovskogo, 48,
677000, Yakutsk, Russia
Аннотация:
In 1940, in attempts to solve the Four Color Problem, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P5 of 3-polytopes with minimum degree 5.
Given a 3-polytope P, by w(P) (h(P)) we denote the minimum degree-sum (minimum of the maximum degrees) of the neighborhoods of 5-vertices in P.
A 5∗-vertex is a 5-vertex adjacent to four 5-vertices. It is known that if a polytope P in P5 has a 5∗-vertex, then h(P) can be arbitrarily large.
For each P without vertices of degrees from 6 to 9 and 5∗-vertices in P5, it follows from Lebesgue's Theorem that w(P)⩽44 and h(P)⩽14.
In this paper, we prove that every such polytope P satisfies w(P)⩽42 and h(P)⩽12, where both bounds are tight.
Ключевые слова:
planar map, planar graph, 3-polytope, structural properties, height, weight.
Поступила 18 мая 2016 г., опубликована 30 июня 2016 г.
Образец цитирования:
O. V. Borodin, A. O. Ivanova, “Light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5”, Сиб. электрон. матем. изв., 13 (2016), 584–591
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/semr695 https://www.mathnet.ru/rus/semr/v13/p584
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