Geodesic
Light Minor 5-Stars in 3-Polytopes with Minimum Degree 5 and No 6-Vertices
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 985-994

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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 w(P) denote the minimum of the degree-sum (weight) of the neighborhoods of 5-vertices (minor 5-stars) in P. In 1996, Jendrol’ and Madaras showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to four 5-vertices, then w(P) can be arbitrarily large. For each P in P5 without vertices of degree 6 and 5-vertices adjacent to four 5-vertices, it follows from Lebesgue’s Theorem that w(P) ≤ 68. Recently, this bound was lowered to w(P) ≤ 55 by Borodin, Ivanova, and Jensen and then to w(P) ≤ 51 by Borodin and Ivanova. In this note, we prove that every such polytope P satisfies w(P) ≤ 44, which bound is sharp.
Keywords: planar map, planar graph, 3-polytope, structural properties, 5-star, weight, height 
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Borodin, Oleg V.; Ivanova, Anna O.; Vasil’eva, Ekaterina I. Light Minor 5-Stars in 3-Polytopes with Minimum Degree 5 and No 6-Vertices. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 4, pp. 985-994. http://geodesic.mathdoc.fr/item/DMGT_2020_40_4_a2/