Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs: Strengthening Conjectures and Proving Paths

Core Concepts

The author strengthens the conjecture of crossing-free Hamiltonian cycles and paths in simple drawings, proving them for various classes including strongly c-monotone and cylindrical drawings.

Abstract

The content discusses the conjecture that every simple drawing of a complete graph contains a crossing-free Hamiltonian cycle. It explores different classes of simple drawings, such as x-monotone, x-bounded, and cylindrical drawings. The author proves the existence of crossing-free Hamiltonian paths between all pairs of vertices in strongly c-monotone and cylindrical drawings.
The paper also establishes a connection between Conjectures 1.1 and 1.2, showing that if Conjecture 1.2 holds for all simple drawings of Kn+1, then Conjecture 1.1 is true for all simple drawings of Kn. Various observations and propositions are made to support these claims, including the verification of the conjectures for specific sub-classes like x-monotone and strongly c-monotone drawings.
Overall, the content delves into the intricacies of crossing-free Hamiltonian cycles and paths in different types of simple graph drawings, providing insights into their existence and properties across various classes.

Stats

For n ≤ 7 up to weak isomorphism.
Lower bound Ω(log(n)1/6) for longest crossing-free path.
Lower bound Ω(√n) for largest plane matching.
Lower bound Ω(log(n)/ log(log(n))) for longest crossing-free path.
Lower bound Ω(log(n)/ log(log(n))) for longest crossing-free cycle.
Existence of completely uncrossed edges in certain classes like x-bounded and cylindrical drawings.

Quotes

"Every simple drawing contains at least one crossing-free Hamiltonian cycle." - Rafla [32]
"Conjecture 1.2 implies a positive answer to Conjecture 1.1." - Author

Key Insights Distilled From

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

by Oswin Aichho... at arxiv.org 03-05-2024

https://arxiv.org/pdf/2303.15610.pdf

Deeper Inquiries

How do generalized twisted drawings impact the existence of crossing-free paths

Generalized twisted drawings play a crucial role in determining the existence of crossing-free paths. In a generalized twisted drawing, there is a specific order on the vertices such that two edges cross if and only if they are nested. This unique property allows for the creation of crossing-free paths within these drawings. By utilizing this characteristic, it becomes possible to navigate through the drawing without encountering any crossings, thus facilitating the establishment of crossing-free paths between different pairs of vertices.

What implications do completely uncrossed edges have on proving conjectures

Completely uncrossed edges have significant implications when proving conjectures related to Hamiltonian paths in simple drawings. These edges serve as key components that can be added to existing sub-drawings without introducing new crossings. The presence of completely uncrossed edges enables the construction of crossing-free paths and cycles within the drawing, providing essential building blocks for establishing conjectures regarding Hamiltonian cycles and paths. Their absence may complicate or hinder efforts to prove certain conjectures due to limitations in creating uninterrupted pathways between vertices.

How does the concept of gap edges contribute to understanding Hamiltonian paths

The concept of gap edges plays a vital role in understanding Hamiltonian paths within strongly c-monotone drawings. In these drawings, gap edges are associated with gaps formed by consecutive vertices around an origin point O. If all gap edges are contained within their respective gaps, then it indicates that no edge crosses over into another region defined by a gap edge's wedge area. This property ensures that there are clear pathways along gap edges that can be utilized to construct crossing-free Hamiltonian paths between specified pairs of vertices within the strongly c-monotone drawing.