Let \( w_{k} \) 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 \( w_{1}\leq 5 \). Back in 1955, Kotzig proved that \( w_{2}\leq 13 \) (so there is an edge of weight at most 13), which is sharp. In 1993, Ando, Iwasaki, and Kaneko proved that \( w_{3}\leq 21 \), which is also sharp due to a construction by Jendrol’ of 1997. In 1997, Borodin refined this by proving that \( w_{3}\leq 18 \) for 3-polytopes with \( w_{2}\geq 7 \), while \( w_{3}\leq 17 \) holds for 3-polytopes with \( w_{2}\geq 8 \), 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 \( K_{4}-e \)). It is known that such 3-polytopes satisfy \( w_{1}\leq 4 \); and, moreover, \( w_{2}\leq 9 \) 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 \( w_{3}\leq 15 \), which is sharp.

令\( w_{k} \)为 3-多胞体中\( k \) -顶点路径 ( \( k \) -paths)中顶点的最小度和（权重）的最大值。简单地说，每个 3 多面体都有一个度数最多为 5 的顶点，因此\( w_{1}\leq 5 \)。早在 1955 年，Kotzig 就证明了\( w_{2}\leq 13 \)（因此存在权重最多为 13 的边缘），这是尖锐的。 1993 年，Ando、Iwasaki 和 Kaneko 证明了\( w_{3}\leq 21 \)，由于 Jendrol' 1997 年的构造，它也是尖锐的。1997 年，Borodin 通过证明\( w_{ 3}\leq 18 \)对于具有\( w_{2}\geq 7 \)的 3-多胞形，而\( w_{3}\leq 17 \)对于具有\( w_{2}\geq的 3-多胞形8 \)，其中 18 的清晰度得到了 Borodin 等人的证实。 2013年，17岁的事情早就知道了。在过去的三十年中，大量研究致力于在这种或那种意义上稀疏的平面图上的结构和着色问题。在本文中，我们处理没有相邻 3 周期的 3 多胞体，即没有弦 4 周期（换句话说，没有\( K_{4}-e \)）。已知这样的3-多胞形满足\(w_{1}\leq 4 \)；而且，\( w_{2}\leq 9 \)成立，其中两个界限都是尖锐的（Borodin，1992）。我们现在证明，每个没有弦 4 循环的 3 多胞体都有一个权重最多为 15 的 3 路径；所以\( w_{3}\leq 15 \)，这是锋利的。