Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs

TL;DR

This paper explores the existence of crossing-free Hamiltonian paths in simple drawings of complete graphs, proving the conjecture for certain classes and analyzing relationships among different drawing classes.

Contribution

It strengthens the longstanding conjecture to include Hamiltonian paths and verifies it for specific classes like strongly c-monotone and cylindrical drawings.

Findings

The stronger conjecture holds for strongly c-monotone drawings.

The stronger conjecture holds for cylindrical drawings.

Provides an overview of classes of simple drawings and their inclusion relations.

Abstract

It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each pair of vertices" and show that this stronger conjecture holds for several classes of simple drawings, including strongly c-monotone drawings and cylindrical drawings. As a second main contribution, we give an overview on different classes of simple drawings and investigate inclusion relations between them up to weak isomorphism.