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Plane Hamiltonian Cycles in Convex Drawings

Authors Helena Bergold , Stefan Felsner , Meghana M. Reddy , Joachim Orthaber , Manfred Scheucher

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Author Details

Helena Bergold
  • Institut für Informatik, Freie Universität Berlin, Germany
Stefan Felsner
  • Institut für Mathematik, Technische Universität Berlin, Germany
Meghana M. Reddy
  • Department of Computer Science, ETH Zürich, Switzerland
Joachim Orthaber
  • Institute of Software Technology, Graz University of Technology, Austria
Manfred Scheucher
  • Institut für Mathematik, Technische Universität Berlin, Germany

Funding

  • Bergold, Helena: Supported by DFG Research Training Group "Facets of Complexity" (DFG-GRK 2434).
  • Felsner, Stefan: Partially supported by DFG Grant FE 340/13-1.
  • M. Reddy, Meghana: Supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681.
  • Orthaber, Joachim: Supported by the Austrian Science Fund (FWF) grant W1230.
  • Scheucher, Manfred: Supported by DFG Grant SCHE 2214/1-1.

Cite AsGet BibTex

Helena Bergold, Stefan Felsner, Meghana M. Reddy, Joachim Orthaber, and Manfred Scheucher. Plane Hamiltonian Cycles in Convex Drawings. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.18

Abstract

A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph K_n admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length Ω(log n / log log n) and of a plane matching of size Ω(n^{1/2}) in every simple drawing of K_n. Instead of studying simpler substructures, we prove Rafla’s conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of K_n contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane k-cycle for each 3 ≤ k ≤ n (pancyclicity), and present further results on maximal plane subdrawings.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Theory of computation → Computational geometry
Keywords
  • simple drawing
  • convexity hierarchy
  • plane pancyclicity
  • plane Hamiltonian connectivity
  • maximal plane subdrawing

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References

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