A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph $K_n$ admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length $\Omega(\log n / \log \log n)$ and of a plane matching of size $\Omega(n^{1/2})$ in every simple drawing of $K_{n}$. Instead of studying simpler substructures, we prove Rafla's conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of $K_n$ contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane $k$-cycle for each $3 \leq k \leq n$ (pancyclicity), and present further results on maximal plane subdrawings.

翻译：Rafla于1988年提出的猜想指出，完全图$K_n$的任意简单绘制均存在一个平面哈密顿圈。事实证明，即使证明此类绘制中存在更简单的非交叉子结构也十分困难。近期研究取得了进展：Aichholzer等人以及Suk和Zeng分别证明了在$K_n$的任意简单绘制中，存在长度为$\Omega(\log n / \log \log n)$的平面路径和大小为$\Omega(n^{1/2})$的平面匹配。不同于研究更简单的子结构，我们证明了Rafla猜想对凸绘制子类成立——这是Arroyo等人引入的凸性层级中最一般的类别。此外，我们证明了$K_n$的每个凸绘制在每对顶点间包含一条平面哈密顿路径（哈密顿连通性），并对每个$3 \leq k \leq n$包含一个平面$k$-圈（泛圈性），同时给出了关于最大平面子绘制的进一步结论。