Almost all about light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5

Abstract Let w Δ be the minimum integer W with the property that every 3-polytope with minimum degree 5 and maximum degree Δ has a vertex of degree 5 with the degree-sum (weight) of all vertices in its neighborhood at most W. Trivially, w 5 = 30 and w 6 = 35 . In 1940, Lebesgue proved w Δ ≤ Δ + 31 for all Δ ≥ 5 and w Δ ≤ Δ + 27 for Δ ≥ 41 . In 1998, the first Lebesgue's result was improved by Borodin and Woodall to w Δ ≤ Δ + 30 . This bound is sharp for Δ = 7 due to Borodin (1992) and Jendrol' and Madaras (1996), Δ = 9 due to Borodin and Ivanova (2013), Δ = 10 due to Jendrol' and Madaras (1996), and Δ = 12 due to Borodin and Woodall (1998). As for the second Lebesgue's bound, Borodin, Ivanova, and Jensen (2014) proved that w Δ = Δ + 27 for Δ ≥ 28 , but w 20 ≥ 48 ; the former fact was extended by Borodin and Ivanova (2016) to w Δ = Δ + 27 for Δ ≥ 24 . In 2017, we proved w Δ ≤ Δ + 29 whenever Δ ≥ 13 , and showed by constructions that w 8 = 38 , w 11 = 41 , and w 13 = 42 . Li, Rao, and Wang (2019) proved w Δ ≤ Δ + 28 for Δ ≥ 16 . The purpose of this paper is to prove that w Δ = Δ + 28 whenever 14 ≤ Δ ≤ 20 . Thus w Δ remains unknown only for 21 ≤ Δ ≤ 23 .