Abstract: Let w be the minimum integer W with the property that every 3-polytope with minimum degree 5 and maximum degree has a vertex of degree 5 with the degreesum (weight) of all vertices in its neighborhood at most W. Trivially, w5 = 30 and w6 = 35. In 1940, Lebesgue proved w ≤ + 31 for all ≥ 5 and w ≤ + 27 for ≥ 41. In 1998, the first Lebesgue’s result was improved by Borodin and Woodall to w ≤ +30. This bound is sharp for = 7 due to Borodin (1992) and Jendrol’ and Madaras (1996), = 9 due to Borodin and Ivanova (2013), = 10 due to Jendrol’ and Madaras (1996), and = 12 due to Borodin and Woodall (1998). As for the second Lebesgue’s bound, Borodin, Ivanova, and Jensen (2014) proved that w = + 27 for ≥ 28, but w20 ≥ 48; the former fact was extended by Borodin and Ivanova (2016) to w = + 27 for ≥ 24. In 2017, we proved w ≤ + 29 whenever ≥ 13, and showed by constructions that w8 = 38, w11 = 41, and w13 = 42. Li, Rao, and Wang (2019) proved w ≤ + 28 for ≥ 16. The purpose of this paper is to prove that w = + 28 whenever 14 ≤ ≤ 20. Thus w remains unknown only for 21 ≤ ≤ 23.

Cite: Borodin O.V. , Ivanova A.O.
Almost all about light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5
Discrete Mathematics. 2022. V.345. N2. P.112678. DOI: 10.1016/j.disc.2021.112678 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Oct 4, 2021
Accepted:	Oct 5, 2021
Published online:	Oct 18, 2021

Identifiers:

Web of science:	WOS:000710162800002
Scopus:	2-s2.0-85117236787
Elibrary:	47513820
OpenAlex:	W3206986005

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