Using SAT to study plane Hamiltonian substructures in simple drawings

Helena Bergold; Stefan Felsner; Meghana M. Reddy; Manfred Scheucher

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arXiv · 2305.09432

Using SAT to study plane Hamiltonian substructures in simple drawings

Helena Bergold·Stefan Felsner·Meghana M. Reddy·Manfred Scheucher

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Abstract

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.

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Helena Bergold, Stefan Felsner, Meghana M. Reddy, Manfred Scheucher. 2023-05-16. Using SAT to study plane Hamiltonian substructures in simple drawings. https://arxiv.org/abs/2305.09432

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