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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 farfrom being resolved. The lower bounds for plane paths and plane matchings haverecently been raised to $(\log n)^{1-o(1)}$ and $\Omega(\sqrt{n})$,respectively. We develop a SAT framework which allows the study of simpledrawings of $K_n$. Based on the computational data we conjecture that everysimple drawing of $K_n$ contains a plane Hamiltonian subgraph with $2n-3$edges. We prove this strengthening of Rafla's conjecture for convex drawings, arich subclass of simple drawings. Our computer experiments also led to othernew challenging conjectures regarding plane substructures in simple drawings ofcomplete graphs.

Using SAT to study plane Hamiltonian substructures in simple drawings