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).

3-polytope 中的3-path \( uvw \)是\( (i,j,k) \) -path if \( d(u)\leq i \) , \( d(v)\leq j \)和\( d(w)\leq k \)，其中 \( d(x) \)是顶点\( x \)的度数 。众所周知，每个 3-polytope 都有一个度数最多为 5 的顶点，称为次要的。如果 3-polytope 中的 3-paths 的每个三元组的中心项至少为 6，则它是次要或主要的描述。早在 1922 年，富兰克林就证明了每个具有最小阶数 5 的 3-polytope 都有一个 \( (6, 5,6) \) -描述严密的路径。在 2016 年，我们证明了每个最小度数为 5 的多面体都有一个 \( (5,6,6) \)-路径也很紧。对于任意的 3-polytopes，Jendrol' (1996) 给出了以下 3-path 的描述：(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),但不知道描述是否严密。Borodin 等人在 2013 年获得了对 3 路径的第一个严格描述：(3,4,11), (3,7,5), (3,10,4), (3,15,3), (4,4,9), (6,4,8), (7,4,7), (6,5,6)。Borodin、Ivanova 和 Kostochka 在 2017 年给出了另一个严格的描述：(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) 本文的目的是获得以下主要的严格描述任意 3-polytopes 的 3-paths: (3,18,3),(3,11,4),(3,8,5),(3,7,6),(4,9,4),( 4,7,5),(5,6,6)。