Using SAT to study plane Hamiltonian substructures in simple drawings
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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 far from being resolved. The lower bounds for plane paths and plane matchings have recently been raised to (log n)^1-o(1) and Ω(√(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.
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