Abstract: The weight w(e) of an edge e in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge e = uv is an (i, j)-edge if d(u) ≤ i and d(v) ≤ j. In 1940, Lebesgue proved that every NPM has a (3, 11)-edge, or (4, 7)-edge, or (5, 6)-edge, where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-polytope has an edge e with w(e) ≤ 13, which bound is sharp. Borodin (1987), answering Erdo os’ question, proved that every NPM has such an edge. Moreover, Borodin (1991) re ned this by proving that there is either a (3, 10)-edge, or (4, 7)-edge, or (5, 6)-edge. Given an NPM, we observe some upper bounds on the minimum weight of all its edges, denoted by w, of those incident with a 3-face, w∗, and those incident with two 3-faces, w∗∗. In particular, Borodin (1996) proved that if w∗∗ = ∞, that is if an NPM has no edges incident with two 3-faces, then either w∗ ≤ 9 or w ≤ 8, where both bounds are sharp. The purpose of our note is to re ne this result by proving that in fact w∗∗ = ∞ implies either a (3, 6)- or (4, 4)-edge incident with a 3-face, or a (3, 5)-edge, which description is tight.

Cite: Borodin O.V. , Ivanova A.O.
Describing edges in normal plane maps having no adjacent 3-faces
Сибирские электронные математические известия (Siberian Electronic Mathematical Reports). 2024. V.21. N1. P.495-500. DOI: 10.33048/semi.2024.21.035 WOS Scopus

Dates:

Submitted:	Nov 14, 2023
Published print:	Jun 23, 2024
Published online:	Jun 23, 2024

Identifiers:

Web of science:	WOS:001283159700012
Scopus:	2-s2.0-85204441730

Citing: Пока нет цитирований

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