In 1988 Rafla conjectured that every simple drawing of a complete graph $K_n$ contains a plane, i.e., non-crossing, Hamiltonian cycle. The conjecture is far from being resolved. The lower bounds for plane paths and plane matchings have recently been raised to $(\log n)^{1-o(1)}$ and $\Omega(\sqrt{n})$, respectively. We develop a SAT framework which allows the study of simple drawings of $K_n$. Based on the computational data we conjecture that every simple drawing of $K_n$ contains a plane Hamiltonian subgraph with $2n-3$ edges. We prove this strengthening of Rafla's conjecture for convex drawings, a rich subclass of simple drawings. Our computer experiments also led to other new challenging conjectures regarding plane substructures in simple drawings of complete graphs.

翻译：1988年，Rafla猜想：完全图$K_n$的每个简单绘图都包含一个平面（即无交叉）哈密顿圈。该猜想至今尚未解决。平面路径和平面匹配的下界最近分别被提升至$(\log n)^{1-o(1)}$和$\Omega(\sqrt{n})$。我们开发了一个SAT框架，用于研究$K_n$的简单绘图。基于计算数据，我们猜想：$K_n$的每个简单绘图都包含一个具有$2n-3$条边的平面哈密顿子图。我们针对简单绘图的一个丰富子类——凸绘图，证明了这一对Rafla猜想的加强形式。我们的计算机实验还引发了关于完全图简单绘图中平面子结构的其他具有挑战性的新猜想。