Crossing-Free Hamiltonian Cycles in Simple Drawings of the Complete Graph: (And What We Found Along the Way)

We study certain representations of graphs in the plane and on the sphere. In particular, we consider simple drawings (where two edges are allowed to have at most one point in common) and straight-line drawings (where all edges are straight line segments) of complete graphs.
One main focus lies on the conjecture that every simple drawing of every complete graph contains a crossing-free Hamiltonian cycle (Rafla’s conjecture). We extend the set of simple drawings where we can show the conjecture to be true and give an overview on approaches that do not work to prove the conjecture in general. Moreover, we also state and investigate a more general conjecture.
In the process, we study properties of and relations between certain subclasses of simple drawings (for example, x-monotone drawings, x-bounded drawings, cylindrical drawings, and c-monotone drawings).
Furthermore, we focus on counting crossing-free Hamiltonian cycles. On the one hand, we consider the optimal crossing-free Hamiltonian cycle problem which asks for the maximum number of different crossing-free Hamiltonian cycles that a single simple drawing or straight-line drawing of a complete graph can contain. Especially, we give some intuition on how the answer to this question might look like.
On the other hand, we initiate research into the minimum (in some broader sense) number of crossing-free Hamiltonian cycles in simple drawings of complete graphs. Our results in this direction also give some indication on why Rafla’s conjecture might be true in general.
To round off the thesis, we give a quick overview on some further topics that we came across in the process of working on the before mentioned ones. More specifically, we introduce what we call ice-cream cone drawings of simple drawings, we reprove a result on empty triangles in simple drawings, and we initiate the study of “generating all possible crossings” with as few simple or straight-line drawings as possible.