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A Solution to Ringel’s Circle Problem

Authors James Davies, Chaya Keller , Linda Kleist , Shakhar Smorodinsky , Bartosz Walczak

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • circle arrangement
  • chromatic number
  • Gallai’s theorem
  • polynomial method

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Abstract

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.

Cite As Get BibTex

James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak. A Solution to Ringel’s Circle Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.33

Author Details

James Davies
  • University of Waterloo, Canada
Chaya Keller
  • Ariel University, Israel
Linda Kleist
  • Technische Universität Braunschweig, Germany
Shakhar Smorodinsky
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Bartosz Walczak
  • Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

Funding

  • Keller, Chaya: Research partially supported by the Israel Science Foundation (grant no. 1065/20).
  • Smorodinsky, Shakhar: Research partially supported by the Israel Science Foundation (grant no. 1065/20).
  • Walczak, Bartosz: The author is partially supported by the National Science Center of Poland grant 2019/34/E/ST6/00443.

Acknowledgements

This work was initiated at the online workshop "Geometric graphs and hypergraphs". We thank the organizers Torsten Ueckerdt and Yelena Yuditsky for a very nice workshop and all participants for fun coffee breaks and a fruitful atmosphere.

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