Stefan Felsner

Helena Bergold, Stefan Felsner, Meghana M. Reddy et al.

A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph $K_n$ admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length $Ω(\log n / \log \log n)$ and of a plane matching of size $Ω(n^{1/2})$ in every simple drawing of $K_n$. Instead of studying simpler substructures, we prove Rafla's conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of $K_n$ contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane $k$-cycle for each $3 \leq k \leq n$ (pancyclicity), and present further results on maximal plane subdrawings.

Timo Brand, Stefan Felsner, Henry Förster et al.

We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.