--- _id: '9295' abstract: - lang: eng text: "Hill's Conjecture states that the crossing number cr(\U0001D43E\U0001D45B) \ of the complete graph \U0001D43E\U0001D45B in the plane (equivalently, the sphere) is 14⌊\U0001D45B2⌋⌊\U0001D45B−12⌋⌊\U0001D45B−22⌋⌊\U0001D45B−32⌋=\U0001D45B4/64+\U0001D442(\U0001D45B3) . 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 \ \U0001D45B4/64+\U0001D442(\U0001D45B3) , thus matching asymptotically the conjectured value of cr(\U0001D43E\U0001D45B) . Let cr\U0001D443(\U0001D43A) denote the crossing number of a graph \U0001D43A in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of \U0001D43E\U0001D45B is (\U0001D45B4/8\U0001D70B2)+\U0001D442(\U0001D45B3) . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if lim\U0001D45B→∞ cr\U0001D443(\U0001D43E\U0001D45B)/\U0001D45B4=1/8\U0001D70B2 . We construct drawings of \U0001D43E\U0001D45B in the projective plane that disprove this." acknowledgement: "We thank two reviewers for their corrections and suggestions on the original version of this\r\npaper. 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 SkłodowskaCurie grant agreement No 754411, IST, Klosterneuburg, Austria." article_processing_charge: No article_type: original author: - first_name: Alan M full_name: Arroyo Guevara, Alan M id: 3207FDC6-F248-11E8-B48F-1D18A9856A87 last_name: Arroyo Guevara orcid: 0000-0003-2401-8670 - first_name: Dan full_name: Mcquillan, Dan last_name: Mcquillan - first_name: R. Bruce full_name: Richter, R. Bruce last_name: Richter - first_name: Gelasio full_name: Salazar, Gelasio last_name: Salazar - first_name: Matthew full_name: Sullivan, Matthew last_name: Sullivan citation: ama: Arroyo Guevara AM, Mcquillan D, Richter RB, Salazar G, Sullivan M. Drawings of complete graphs in the projective plane. Journal of Graph Theory. 2021;97(3):426-440. doi:10.1002/jgt.22665 apa: 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 chicago: Arroyo Guevara, Alan M, Dan Mcquillan, R. Bruce Richter, Gelasio Salazar, and Matthew Sullivan. “Drawings of Complete Graphs in the Projective Plane.” Journal of Graph Theory. Wiley, 2021. https://doi.org/10.1002/jgt.22665. ieee: A. M. Arroyo Guevara, D. Mcquillan, R. B. Richter, G. Salazar, and M. Sullivan, “Drawings of complete graphs in the projective plane,” Journal of Graph Theory, vol. 97, no. 3. Wiley, pp. 426–440, 2021. ista: Arroyo Guevara AM, Mcquillan D, Richter RB, Salazar G, Sullivan M. 2021. Drawings of complete graphs in the projective plane. Journal of Graph Theory. 97(3), 426–440. mla: Arroyo Guevara, Alan M., et al. “Drawings of Complete Graphs in the Projective Plane.” Journal of Graph Theory, vol. 97, no. 3, Wiley, 2021, pp. 426–40, doi:10.1002/jgt.22665. short: A.M. Arroyo Guevara, D. Mcquillan, R.B. Richter, G. Salazar, M. Sullivan, Journal of Graph Theory 97 (2021) 426–440. date_created: 2021-03-28T22:01:41Z date_published: 2021-03-23T00:00:00Z date_updated: 2023-08-07T14:26:15Z day: '23' department: - _id: UlWa doi: 10.1002/jgt.22665 ec_funded: 1 external_id: arxiv: - '2002.02287' isi: - '000631693200001' intvolume: ' 97' isi: 1 issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2002.02287 month: '03' oa: 1 oa_version: Preprint page: 426-440 project: - _id: 260C2330-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '754411' name: ISTplus - Postdoctoral Fellowships publication: Journal of Graph Theory publication_identifier: eissn: - 1097-0118 issn: - 0364-9024 publication_status: published publisher: Wiley quality_controlled: '1' scopus_import: '1' status: public title: Drawings of complete graphs in the projective plane type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 97 year: '2021' ...