Describing 3-faces in 3-polytopes without adjacent triangles Full article
| Journal |
Siberian Mathematical Journal
ISSN: 0037-4466 , E-ISSN: 1573-9260 |
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| Output data | Year: 2025, Volume: 66, Number: 1, Pages: 16-21 Pages count : 6 DOI: 10.1134/S0037446625010021 | ||||
| Tags | plane graph, 3-polytope, sparse 3-polytope, structural property, 3-face, weight | ||||
| Authors |
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| Affiliations |
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Funding (2)
| 1 | Sobolev Institute of Mathematics | FWNF-2022-0017 |
| 2 | Министерство науки и высшего образования РФ | FSRG-2023-0025 |
Abstract:
Over the last several decades, much research has been devoted to structural and coloring problems on the plane graphs that are sparse in some sense. In this paper we deal with the densest instances of sparse 3-polytopes, namely, those without adjacent 3-cycles. Borodin proved in 1996 that such 3-polytope has a vertex of degree at most 4 and, moreover, an edge with the degree-sum of its end-vertices at most 9, where both bounds are sharp. Denote the degree of a vertex v by d(v). An edge e = xy in a 3-polytope is an (i, j)-edge if d(x) ≤ i and d(y) ≤ j. The well-known (3,5;4,4)-Archimedean solid corresponds to a plane quadrangulation in which every edge joins a 3-vertex with a 5-vertex. In particular, this 3-polytope has no 3-cycles. Recently, Borodin and Ivanova proved that every 3-polytope with neither adjacent 3-cycles nor (3, 5)-edges has a 3-face with the degree-sum of its incident vertices (weight) at most 16, which bound is sharp. A 3-face f = (x, y, z) is an (i, j, k)-face or a face of type (i, j, k) if d(x) ≤ i, d(y) ≤ j, and d(z) ≤ k. The purpose of this paper is to prove that there are precisely two tight descriptions of 3-face-types in 3-polytopes without adjacent 3-cycles under the above-mentioned necessary assumption of the absence of (3, 5)-edges; namely, {(3, 6, 7) ∨ (4, 4, 7)} and {(4, 6, 7)}. This implies that there is a unique tight description of 3-faces in 3-polytopes with neither adjacent 3-cycles nor 3-vertices: {(4, 4, 7)}.
Cite:
Borodin O.V.
, Ivanova A.O.
Describing 3-faces in 3-polytopes without adjacent triangles
Siberian Mathematical Journal. 2025. V.66. N1. P.16-21. DOI: 10.1134/S0037446625010021 WOS Scopus РИНЦ OpenAlex
Describing 3-faces in 3-polytopes without adjacent triangles
Siberian Mathematical Journal. 2025. V.66. N1. P.16-21. DOI: 10.1134/S0037446625010021 WOS Scopus РИНЦ OpenAlex
Original:
Бородин О.В.
, Иванова А.О.
Описание 3–граней в 3–многогранниках без смежных треугольников
Сибирский математический журнал. 2025. Т.66. №1. С.20-26. DOI: 10.33048/smzh.2025.66.102 РИНЦ
Описание 3–граней в 3–многогранниках без смежных треугольников
Сибирский математический журнал. 2025. Т.66. №1. С.20-26. DOI: 10.33048/smzh.2025.66.102 РИНЦ
Dates:
| Submitted: | Oct 30, 2024 |
| Accepted: | Dec 25, 2024 |
| Published print: | Jan 8, 2025 |
| Published online: | Jan 8, 2025 |
Identifiers:
| Web of science: | WOS:001410783000018 |
| Scopus: | 2-s2.0-85217445467 |
| Elibrary: | 80239644 |
| OpenAlex: | W4406946719 |
Citing:
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