Реферат: Attempting to solve the Four Color Problem in 1940, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P 5 of 3-polytopes with minimum degree 5. This description depends on 32 main parameters. Not many precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in P 5 . Given a 3-polytope P, by w(P) denote the minimum of the maximum degree-sum (weight) of the neighborhoods of 5-vertices (minor 5-stars) in P. In 1996, Jendrol’ and Madaras showed that if a polytope P in P 5 is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5, ∞)-star), then w(P) can be arbitrarily large. For each P* in P 5 with neither vertices of degree 6 and 7 nor minor (5, 5, 5, 5, ∞)-star, it follows from Lebesgue’s Theorem that w(P*) ≤ 51. We prove that every such polytope P* satisfies w(P*) ≤ 42, which bound is sharp. This result is also best possible in the sense that if 6-vertices are allowed but 7-vertices forbidden, or vice versa; then the weight of all minor 5-stars in P 5 under the absence of minor (5, 5, 5, 5, ∞)-stars can reach 43 or 44, respectively. © 2019, Pleiades Publishing, Ltd.

Библиографическая ссылка: Borodin O.V. , Ivanova A.O.
Light Minor 5-Stars in 3-Polytopes with Minimum Degree 5
Siberian Mathematical Journal. 2019. V.60. N2. P.272-278. DOI: 10.1134/S0037446619020071 WOS Scopus OpenAlex

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Web of science:	WOS:000465640100007
Scopus:	2-s2.0-85064896692
OpenAlex:	W2942032962

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Scopus	1
Web of science	1

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