Abstract
A 3-path \( uvw \) in a 3-polytope is an \( (i,j,k) \)-path if \( d(u)\leq i \), \( d(v)\leq j \), and \( d(w)\leq k \), where \( d(x) \) is the degree of a vertex \( x \). It is well known that each 3-polytope has a vertex of degree at most 5 called minor. A description of 3-paths in a 3-polytope is minor or major if the central item of its every triplet is at least 6. Back in 1922, Franklin proved that each 3-polytope with minimum degree 5 has a \( (6,5,6) \)-path which description is tight. In 2016, we proved that each polytope with minimum degree 5 has a \( (5,6,6) \)-path which is also tight. For arbitrary 3-polytopes, Jendrol’ (1996) gave the following description of 3-paths: (10,3,10), (7,4,7),(6,5,6),(3,4,15),(3,6,11),(3,8,5),(3,10,3),(4,4,11),(4,5,7),(4,7,5), but it is unknown whether the description is tight or not. The first tight description of 3-paths was obtained in 2013 by Borodin et al.: (3,4,11), (3,7,5), (3,10,4), (3,15,3), (4,4,9), (6,4,8), (7,4,7), (6,5,6). Another tight description was given by Borodin, Ivanova, and Kostochka in 2017: (3,15,3), (3,10,4), (3,8,5), (4,7,4), (5,5,7), (6,5,6), (3,4,11), (4,4,9), (6,4,7) The purpose of this paper is to obtain the following major tight descriptions of 3-paths for arbitrary 3-polytopes: (3,18,3),(3,11,4),(3,8,5),(3,7,6),(4,9,4),(4,7,5),(5,6,6).
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This work was funded by the Russian Science Foundation (Grant 16–11–10054).
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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 3, pp. 502–512. https://doi.org/10.33048/smzh.2021.62.302
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Borodin, O.V., Ivanova, A.O. A Tight Description of 3-Polytopes by Their Major 3-Paths. Sib Math J 62, 400–408 (2021). https://doi.org/10.1134/S0037446621030022
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DOI: https://doi.org/10.1134/S0037446621030022