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 $Ω(\log n / \log \log n)$ and of a plane matching of size $Ω(n^{1/2})$ 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：拉夫拉于1988年提出的猜想断言：完全图$K_n$的任意简单绘制均包含一个平面哈密顿圈。事实证明，即使要证明此类绘图中存在更简单的非交叉子结构也极为困难。艾希霍尔泽等人以及Suk和Zeng近期取得进展，他们证明了在$K_n$的每个简单绘制中均存在长度为$Ω(\log n / \log \log n)$的平面路径和规模为$Ω(n^{1/2})$的平面匹配。不同于研究更简单的子结构，我们在凸绘制这一子类中证明了拉夫拉猜想，该子类是阿罗约等人提出的凸性层级中最广义的类别。此外，我们证明了$K_n$的每个凸绘制在任意顶点对间均包含平面哈密顿路径（哈密顿连通性），且对每个$3 \leq k \leq n$均包含平面$k$圈（泛圈性），并进一步给出了关于极大平面子绘制的研究结果。