An Algorithm for Estimating the Crossing Number of Dense Graphs, and Continuous Analogs of the Crossing and Rectilinear Crossing Numbers
Name
454_2025_Article_783.pdf
Size
568.21 KB
Format
Adobe PDF
Checksum (MD5)
9256717622c4f8350fefd95ac84a851d
Author(s)
Solé-Pi, Oriol
Date Issued
October 21, 2025
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Solé-Pi, O. An Algorithm for Estimating the Crossing Number of Dense Graphs, and Continuous Analogs of the Crossing and Rectilinear Crossing Numbers. Discrete Comput Geom (2025).
Version
Final published version
Abstract
We present a deterministic n 2 + o ( 1 ) -time algorithm that approximates the crossing number of any graph G of order n up to an additive error of o ( n 4 ) . We also provide a randomized polynomial-time algorithm that constructs a drawing of G with cr ( G ) + o ( n 4 ) crossings. These results yield a 1 + o ( 1 ) approximation algorithm for the crossing number of dense graphs. Our work complements a paper of Fox, Pach and Súk [20], who obtained similar results for the rectilinear crossing number. The results in [20] and in this paper imply that the (normalized) crossing and rectilinear crossing numbers are estimable parameters. Motivated by this, we introduce two graphon parameters, the crossing density and the rectilinear crossing density, and we prove that, in a precise sense, these are the correct continuous analogs of the crossing and rectilinear crossing numbers of graphs.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00454-025-00783-w
