Heights of minor 5-stars in 3-polytopes with minimum degree 5 and no vertices of degree 6 and 7

Abstract Given a 3-polytope P , by h ( P ) we denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P . In 1940, H.  Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P 5 of 3-polytopes with minimum degree 5. In 1996, S.  Jendrol’ and T.  Madaras showed that if a polytope P in P 5 is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor ( 5 , 5 , 5 , 5 , ∞ ) -star), then h ( P ) can be arbitrarily large. For each P ∗ in P 5 with neither vertices of degree 6 or 7 nor minor ( 5 , 5 , 5 , 5 , ∞ ) -star, it follows from Lebesgue’s theorem that h ( P ∗ ) ≤ 23 . We prove that every such polytope P ∗ satisfies h ( P ∗ ) ≤ 14 , which bound is sharp. Moreover, if 6-vertices are allowed but 7-vertices forbidden, or vice versa, then the height of minor 5-stars in P 5 under the absence of minor ( 5 , 5 , 5 , 5 , ∞ ) -stars can reach 15 or 17, respectively.