We establish new results on the curve complexity of k-colored point-set embeddings when $k=3$. We show that there exist 3-colored caterpillars with only three independent edges whose 3-colored point-set embeddings may require $\mathrm{\Omega }(n^{\frac{1}{3}})$ bends on $\mathrm{\Omega }(n^{\frac{2}{3}})$ edges. This settles an open problem by Badent et al. [5] about the curve complexity of point set embeddings of k-colored trees and it extends a lower bound by Pach and Wenger [35] to the case that the graph only has $O(1)$ independent edges. Concerning upper bounds, we prove that any 3-colored path admits a 3-colored point-set embedding with curve complexity at most 4. In addition, we introduce a variant of the k-colored simultaneous embeddability problem and study its relationship with the k-colored point-set embeddability problem.

我们建立了关于k色点集嵌入的曲线复杂度的新结果$ķ=3$。我们显示存在仅带有三个独立边的3色毛毛虫，它们的3色点集嵌入可能需要$\mathrm{\Omega }（{ñ}^{\frac{\mathrm{1个}}{3}}）$ 弯腰 $\mathrm{\Omega }（{ñ}^{\frac{2}{3}}）$边缘。这解决了Badent等人的公开问题。[5]关于k色树的点集嵌入的曲线复杂度，它扩展了Pach和Wenger [35]的下界，使得图仅具有$Ø（\mathrm{1个}）$独立的边缘。关于上限，我们证明任何3色路径都允许3色点集嵌入，其曲线复杂度最多为4。此外，我们介绍了k色同时嵌入性问题的一种变体，并研究了其与k的关系。彩色点集可嵌入性问题。