Drawings of complete graphs in the projective plane

Arroyo Guevara AM, Mcquillan D, Richter RB, Salazar G, Sullivan M. 2021. Drawings of complete graphs in the projective plane. Journal of Graph Theory.


Journal Article | Epub ahead of print | English

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Author
Arroyo Guevara, Alan MIST Austria; Mcquillan, Dan; Richter, R. Bruce; Salazar, Gelasio; Sullivan, Matthew
Department
Abstract
Hill's Conjecture states that the crossing number cr(𝐾𝑛) of the complete graph 𝐾𝑛 in the plane (equivalently, the sphere) is 14βŒŠπ‘›2βŒ‹βŒŠπ‘›βˆ’12βŒ‹βŒŠπ‘›βˆ’22βŒ‹βŒŠπ‘›βˆ’32βŒ‹=𝑛4/64+𝑂(𝑛3) . Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely 𝑛4/64+𝑂(𝑛3) , thus matching asymptotically the conjectured value of cr(𝐾𝑛) . Let cr𝑃(𝐺) denote the crossing number of a graph 𝐺 in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of 𝐾𝑛 is (𝑛4/8πœ‹2)+𝑂(𝑛3) . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if limπ‘›β†’βˆžβ€Šcr𝑃(𝐾𝑛)/𝑛4=1/8πœ‹2 . We construct drawings of 𝐾𝑛 in the projective plane that disprove this.
Publishing Year
Date Published
2021-03-23
Journal Title
Journal of Graph Theory
Acknowledgement
We thank two reviewers for their corrections and suggestions on the original version of this paper. This project has received funding from NSERC Grant 50503-10940-500 and from the European Union’s Horizon 2020 research and innovation programme under the Marie SkΕ‚odowskaCurie grant agreement No 754411, IST, Klosterneuburg, Austria.
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Cite this

Arroyo Guevara AM, Mcquillan D, Richter RB, Salazar G, Sullivan M. Drawings of complete graphs in the projective plane. Journal of Graph Theory. 2021. doi:10.1002/jgt.22665
Arroyo Guevara, A. M., Mcquillan, D., Richter, R. B., Salazar, G., & Sullivan, M. (2021). Drawings of complete graphs in the projective plane. Journal of Graph Theory. Wiley. https://doi.org/10.1002/jgt.22665
Arroyo Guevara, Alan M, Dan Mcquillan, R. Bruce Richter, Gelasio Salazar, and Matthew Sullivan. β€œDrawings of Complete Graphs in the Projective Plane.” Journal of Graph Theory. Wiley, 2021. https://doi.org/10.1002/jgt.22665.
A. M. Arroyo Guevara, D. Mcquillan, R. B. Richter, G. Salazar, and M. Sullivan, β€œDrawings of complete graphs in the projective plane,” Journal of Graph Theory. Wiley, 2021.
Arroyo Guevara AM, Mcquillan D, Richter RB, Salazar G, Sullivan M. 2021. Drawings of complete graphs in the projective plane. Journal of Graph Theory.
Arroyo Guevara, Alan M., et al. β€œDrawings of Complete Graphs in the Projective Plane.” Journal of Graph Theory, Wiley, 2021, doi:10.1002/jgt.22665.
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