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From Local Pair-Crossing Number to Local Crossing Number

Authors Jacob Fox , János Pach , Andrew Suk

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ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Crossing numbers
  • pair crossing numbers

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Abstract

We prove that if a graph can be drawn in the plane such that each edge crosses at most k other edges, then it can be redrawn so that each edge participates in at most k³+O(k²) crossings. This improves the previous exponential bound that follows from a result of Schaefer and Štefankovič and answers a question of Ackerman and Schaefer.

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Jacob Fox, János Pach, and Andrew Suk. From Local Pair-Crossing Number to Local Crossing Number. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 31:1-31:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.31

Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, CA, USA
János Pach
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

Funding

This manuscript was completed while the authors were in residence at SLMath Berkeley, during the Spring semester of 2025, supported by NSF Grant DMS-1928930.
  • Fox, Jacob: supported by NSF awards DMS-2452737 and DMS-2154129.
  • Pach, János: Rényi Institute, Budapest. Supported by NKFIH grant K-176529, Austrian Science Fund Z 342-N31 and ERC Advanced Grant "GeoScape."
  • Suk, Andrew: Supported by NSF grants DMS-1952786 and DMS-2246847.

Acknowledgements

The authors want to express their gratitude to Géza Tóth for directing their attention to the problem discussed in this note, and for his many valuable remarks and suggestions.

References

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