Light minor 5-stars in 3-polytopes with minimum degree 5 and no 6-vertices 1 Full article
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Discussiones Mathematicae - Graph Theory
ISSN: 1234-3099 , E-ISSN: 2083-5892 |
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| Output data | Year: 2020, Volume: 40, Number: 4, Pages: 985-994 Pages count : 10 DOI: 10.7151/dmgt.2155 | ||
| Tags | 3-polytope; 5-star; Height; Planar graph; Planar map; Structural properties; Weight | ||
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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 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. © 2020 Sciendo. All rights reserved.
Cite:
Borodin O.V.
, Ivanova A.O.
, Vasil'eva E.I.
Light minor 5-stars in 3-polytopes with minimum degree 5 and no 6-vertices 1
Discussiones Mathematicae - Graph Theory. 2020. V.40. N4. P.985-994. DOI: 10.7151/dmgt.2155 WOS Scopus OpenAlex
Light minor 5-stars in 3-polytopes with minimum degree 5 and no 6-vertices 1
Discussiones Mathematicae - Graph Theory. 2020. V.40. N4. P.985-994. DOI: 10.7151/dmgt.2155 WOS Scopus OpenAlex
Identifiers:
| Web of science: | WOS:000551912200003 |
| Scopus: | 2-s2.0-85091461868 |
| OpenAlex: | W2892326853 |