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1-Planar Drawings of Products of Cycles

Authors Stefan Felsner , Kolja Knauer , Yasumi Kogo, Bobby Miraftab

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LIPIcs.GD.2026.31.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • 1-planar
  • drawings
  • cycles
  • product
  • torus

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Abstract

We show that the Cartesian product of two cycles C_m × C_n, i.e. the m × n toroidal grid, admits exactly three distinct simple 1-planar drawings when 5 ≤ m ≤ n. These drawings have (m-2)n, m(n-2), and mn - 4 crossings, respectively. As a corollary, we obtain that the 1-planar crossing number of C_m × C_n with 3 ≤ m ≤ n is (m-2)n. This confirms, in the 1-planar setting, a conjecture of Harary, Kainen, and Schwenk. The drawings have the property that the n fundamental cycles C_m and the m fundamental cycles C_n of C_m× C_n form two classes of pairwise disjoint cycles. In fact, establishing this property is a crucial step in our proof.

Cite As Get BibTex

Stefan Felsner, Kolja Knauer, Yasumi Kogo, and Bobby Miraftab. 1-Planar Drawings of Products of Cycles. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.31

Author Details

Stefan Felsner
  • Institut für Mathematik, Technische Universität Berlin, Germany
Kolja Knauer
  • Departament de Matemàtiques i Informàtica - Universitat de Barcelona, Spain
  • Centre de Recerca Matemàtica, Barcelona, Spain
Yasumi Kogo
  • Institut für Mathematik, Technische Universität Berlin, Germany
Bobby Miraftab
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M), ANR project MIMETIQUE: ANR-25-CE48-4089-01.
  • Knauer, Kolja: supported by grants PID2022-137283NB-C22 funded by MICIU/AEI/10.13039/ 501100011033 and ERDF/EU,

Acknowledgements

This work was started during a research visit of Kolja Knauer in Berlin. Details have been worked out in the master’s thesis of Yasumi Kogo.

References

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