Alapfogalmak
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.
Kivonat
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.
Statisztikák
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.
Idézetek
"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