Rafla conjectured in 1988 that every simple drawing of the complete graph K_n admits a plane Hamiltonian cycle — a cycle visiting every vertex exactly once whose edges don't cross each other. The conjecture has been open for 36 years.
Bergold, Felsner, Reddy, Orthaber, and Scheucher prove it for convex drawings, the largest class in the convexity hierarchy of graph drawings. A convex drawing places vertices so that the convex hull of each face is a triangle — a substantial structural constraint, but one that still allows complex edge crossings.
The proof doesn't just find a Hamiltonian cycle. It establishes Hamiltonian connectivity: plane Hamiltonian paths exist between any pair of vertices. And pancyclicity: plane k-cycles exist for all valid k. The convex drawing structure provides enough rigidity to route edges without crossings, but the routing arguments are nontrivial — the same drawing can have astronomical numbers of crossings among non-adjacent edges.
The full Rafla conjecture — for all simple drawings, not just convex ones — remains open. The difficulty is that general simple drawings can have edges that wind arbitrarily around vertices, making it impossible to argue locally about crossing avoidance. Convex drawings constrain the global topology enough to make the argument work.
Thirty-six years, one major subclass resolved. The result confirms that convexity structure in graph drawings is sufficient for plane Hamiltonicity. Whether it's necessary — whether some non-convex drawing of K_n fails to have a plane Hamiltonian cycle — is the remaining question.