Low 5-Stars at 5-Vertices in 3-Polytopes with Minimum Degree 5 and No Vertices of Degree from 7 to 9

Abstract In 1940, 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 h5(P) we denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P. Recently, Borodin, Ivanova and Jensen showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to two 5-vertices and two more vertices of degree at most 6, called a (5, 5, 6, 6, ∞)-vertex, then h5(P) can be arbitrarily large. Therefore, we consider the subclass P*5 of 3-polytopes in P5 that avoid (5, 5, 6, 6, ∞)-vertices. For each P*in P*5 without vertices of degree from 7 to 9, it follows from Lebesgue’s Theorem that h5(P*) ≤ 17. Recently, this bound was lowered by Borodin, Ivanova, and Kazak to the sharp bound h5(P*) ≤ 15 assuming the absence of vertices of degree from 7 to 11 in P*. In this note, we extend the bound h5(P*) ≤ 15 to all P*s without vertices of degree from 7 to 9.