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. © 2020 Sciendo. All rights reserved.

Cite: Borodin O.V. , Bykov M.A. , Ivanova A.O.
Low 5-stars at 5-vertices in 3-polytopes with minimum degree 5 and no vertices of degree from 7 to 9 1
Discussiones Mathematicae - Graph Theory. 2020. V.40. N4. P.1025-1033. DOI: 10.7151/dmgt.2159 WOS Scopus OpenAlex

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Web of science:	WOS:000551912200006
Scopus:	2-s2.0-85062197282
OpenAlex:	W2894474815

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