Abstract: Let wk be the maximum of the minimum degree-sum (weight) of vertices in k-vertex paths (k-paths) in 3-polytopes. Trivially, each 3-polytope has a vertex of degree at most 5, and so w1 ≤ 5. Back in 1955, Kotzig proved that w2 ≤ 13 (so there is an edge of weight at most 13), which is sharp. In 1993, Ando, Iwasaki, and Kaneko proved that w3 ≤ 21, which is also sharp due to a construction by Jendrol’ of 1997. In 1997, Borodin refined this by proving that w3 ≤ 18 for 3-polytopes with w2 ≥ 7, while w3 ≤ 17 holds for 3-polytopes with w2 ≥ 8, where the sharpness of 18 was confirmed by Borodin et al. in 2013, and that of 17 was known long ago. Over the last three decades, much research has been devoted to structural and coloring problems on the plane graphs that are sparse in this or that sense. In this paper we deal with 3-polytopes without adjacent 3-cycles that is without chordal 4-cycle (in other words, without K4 − e). It is known that such 3-polytopes satisfy w1 ≤ 4; and, moreover, w2 ≤ 9 holds, where both bounds are sharp (Borodin, 1992). We prove now that each 3-polytope without chordal 4-cycles has a 3-path of weight at most 15; and so w3 ≤ 15, which is sharp.

Cite: Borodin O.V. , Ivanova A.O.
Light 3-Paths in 3-Polytopes without Adjacent Triangles
Siberian Mathematical Journal. 2024. V.65. N2. P.257-264. DOI: 10.1134/S0037446624020022 WOS Scopus РИНЦ РИНЦ OpenAlex

Original: Бородин О.В. , Иванова А.О.
Легкие 3-цепи в 3-многогранниках без смежных 3-граней
Сибирский математический журнал. 2024. Т.65. №2. С.249-257. DOI: 10.33048/smzh.2024.65.202 РИНЦ

Dates:

Submitted:	Oct 17, 2023
Accepted:	Nov 28, 2023
Published print:	Mar 25, 2024
Published online:	Mar 25, 2024

Identifiers:

Web of science:	WOS:001256062900002
Scopus:	2-s2.0-85188547892
Elibrary:	66532350 | 67308244
OpenAlex:	W4393167112

Citing: Пока нет цитирований

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