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

In 1940, Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class $P_5$ of 3-polytopes with minimum degree 5. Given a 3-polytope $P$, by $h_5(P)$ 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 $P_5$ 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 $h_5(P)$ can be arbitrarily large. Therefore, we consider the subclass \(P_5^\ast\) of 3-polytopes in $P_5$ that avoid (5, 5, 6, 6, ∞)-vertices. For each $P^\ast$ in $P_5^\ast$ without vertices of degree from 7 to 9, it follows from Lebesgue’s Theorem that $h_5(P^\ast) ≤ 17$. Recently, this bound was lowered by Borodin, Ivanova, and Kazak to the sharp bound $h_5(P^\ast) ≤ 15$ assuming the absence of vertices of degree from 7 to 11 in $P^\ast$. In this note, we extend the bound $h_5(P^\ast) ≤ 15$ to all $P^\ast$s without vertices of degree from 7 to 9.