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

Abstract

Let wΔ 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, w5=30 and w6=35. In 1940, Lebesgue proved wΔ≤Δ+31 for all Δ≥5 and wΔ≤Δ+27 for Δ≥41.In 1998, the first Lebesgue's result was improved by Borodin and Woodall to wΔ≤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 wΔ=Δ+27 for Δ≥28, but w20≥48; the former fact was extended by Borodin and Ivanova (2016) to wΔ=Δ+27 for Δ≥24.In 2017, we proved wΔ≤Δ+29 whenever Δ≥13, and showed by constructions that w8=38, w11=41, and w13=42. Li, Rao, and Wang (2019) proved wΔ≤Δ+28 for Δ≥16.The purpose of this paper is to prove that wΔ=Δ+28 whenever 14≤Δ≤20. Thus wΔ remains unknown only for 21≤Δ≤23.