Using SAT to study plane Hamiltonian substructures in simple drawings

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.