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Abstract
Simple drawings are drawings of graphs in the plane
such that vertices are distinct points in the plane,
edges are Jordan arcs connecting their endpoints, and
edges intersect at most once either in a proper crossing
or in a shared endpoint.
It is conjectured that every simple drawing of the
complete graph with $n$ vertices, $K_n$, contains a plane
Hamiltonian cycle, and consequently a plane Hamil
tonian path. However, to the best of our knowledge,
$\Omega((\log n)^{1/6})$ is currently the best known lower bound
for the length of a plane path contained in any simple
drawing of $K_n$. We improve this bound to $\Omega(\log n / (\log \log n) )$.
such that vertices are distinct points in the plane,
edges are Jordan arcs connecting their endpoints, and
edges intersect at most once either in a proper crossing
or in a shared endpoint.
It is conjectured that every simple drawing of the
complete graph with $n$ vertices, $K_n$, contains a plane
Hamiltonian cycle, and consequently a plane Hamil
tonian path. However, to the best of our knowledge,
$\Omega((\log n)^{1/6})$ is currently the best known lower bound
for the length of a plane path contained in any simple
drawing of $K_n$. We improve this bound to $\Omega(\log n / (\log \log n) )$.
Original language  English 

Pages  44 
Number of pages  1 
Publication status  Published  2021 
Event  XIX Spanish Meeting on Computational Geometry: EGC21  Virtual Madrid, Spain, Virtuell, Spain Duration: 5 Jul 2021 → 7 Jul 2021 https://quantumexplore.com/egc21/ https://quantumexplore.com/egc21program/ 
Conference
Conference  XIX Spanish Meeting on Computational Geometry 

Abbreviated title  EGC21 
Country/Territory  Spain 
City  Virtuell 
Period  5/07/21 → 7/07/21 
Internet address 
Keywords
 Discrete and Computational Geometry
Fields of Expertise
 Information, Communication & Computing
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 1 Active

Doctoral Program: Discrete Mathematics
Ebner, O., Lehner, F., Greinecker, F., Burkard, R., Wallner, J., Elsholtz, C., Woess, W., Raseta, M., Bazarova, A., Krenn, D., Lehner, F., Kang, M., Tichy, R., SavaHuss, E., Klinz, B., Heuberger, C., Grabner, P., Barroero, F., Cuno, J., Kreso, D. & Berkes, I.
1/05/10 → 31/12/22
Project: Research project
Activities

XIX Spanish Meeting on Computational Geometry
Alexandra Weinberger (Participant)
5 Jul 2021 → 7 Jul 2021Activity: Participation in or organisation of › Workshop, seminar or course (Participation in/Organisation of)

Plane paths in simple drawings of complete graphs
Alexandra Weinberger (Speaker)
5 Jul 2021Activity: Talk or presentation › Talk at conference or symposium › Science to science