Abstract: The weight w(e) of an edge e in a 3-polytope is the sum of the degrees 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 3-polytope contains a (3, 11)-, (4, 7)-, or (5, 6)-edge, where 7 and 6 are the best possible. In 1955, Kotzig proved that every 3-polytope contains an edge whose end-vertex degrees sum to at most 13, and this bound is sharp. Borodin (1987), answering a question by Erd˝os, proved that every planar graph without vertices of degree less than 3 contains such an edge. Moreover, Borodin (1991) strengthened this result by proving that there exists either a (3, 10)-, (4, 7)-, or (5, 6)-edge. For 3-polytopes, upper bounds were obtained for the minimal weight (the sum of the degrees of the end vertices) of all its edges, denoted by w; of edges incident to a 3-face, denoted by w∗; and of edges incident to two 3-faces, denoted by w∗∗. In particular, Borodin (1996) proved that if w∗∗ = ∞, i.e., there are no edges incident to two 3-faces, then either w∗ ≤ 9 or w ≤ 8, and both bounds are the best possible. Recently, we have strengthened this fact by proving that w∗∗ = ∞ implies the existence of either a (3, 6)-edge or a (4, 4)-edge incident to a 3-face, or else a (3, 5)-edge, with an exact description. (It is well known that if (3, 5)-edges are present, then 3-faces may be absent altogether.) The aim of our paper is to strengthen the above result by proving that w∗∗ = ∞ implies either a (3, 6)-edge surrounded by a 3-face and a 4-face, or a (4, 4)-edge surrounded by a 3-face and a 7−-face, or a (3, 5)-edge, where none of the parameters can be improved. The main difficulty was to construct a 3-polytope confirming the sharpness of 7 in this description.

Cite: Borodin O.V. , Ivanova A.O.
Description of edges incident to 3-faces in 3-polytopes without adjacent 3-faces
Siberian Mathematical Journal. 2025. V.66. N6. P.1368–1373. DOI: 10.1134/S0037446625060035 Scopus

Original: Бородин О.В. , Иванова А.О.
Описание инцидентных 3–граням ребер в 3–многогранниках без смежных 3–граней
Сибирский математический журнал. 2025. Т.66. №6. С.1030–1036. DOI: 10.33048/smzh.2025.66.603

Dates:

Submitted:	Jun 4, 2025
Accepted:	Aug 15, 2025
Published print:	Nov 19, 2025
Published online:	Nov 19, 2025

Identifiers:

Scopus:

2-s2.0-105022712750

Citing: Пока нет цитирований

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