[{"abstract":[{"lang":"eng","text":"Hill's Conjecture states that the crossing number  cr(𝐾𝑛)  of the complete graph  𝐾𝑛  in the plane (equivalently, the sphere) is  14⌊𝑛2⌋⌊𝑛−12⌋⌊𝑛−22⌋⌊𝑛−32⌋=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  (𝑛4/8𝜋2)+𝑂(𝑛3) . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if  lim𝑛→∞ cr𝑃(𝐾𝑛)/𝑛4=1/8𝜋2 . We construct drawings of  𝐾𝑛  in the projective plane that disprove this."}],"arxiv":1,"date_created":"2021-03-28T22:01:41Z","ec_funded":1,"oa_version":"Preprint","publication":"Journal of Graph Theory","title":"Drawings of complete graphs in the projective plane","department":[{"_id":"UlWa"}],"publisher":"Wiley","citation":{"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.” <i>Journal of Graph Theory</i>, vol. 97, no. 3, Wiley, 2021, pp. 426–40, doi:<a href=\"https://doi.org/10.1002/jgt.22665\">10.1002/jgt.22665</a>.","short":"A.M. Arroyo Guevara, D. Mcquillan, R.B. Richter, G. Salazar, M. Sullivan, Journal of Graph Theory 97 (2021) 426–440.","ama":"Arroyo Guevara AM, Mcquillan D, Richter RB, Salazar G, Sullivan M. Drawings of complete graphs in the projective plane. <i>Journal of Graph Theory</i>. 2021;97(3):426-440. doi:<a href=\"https://doi.org/10.1002/jgt.22665\">10.1002/jgt.22665</a>","apa":"Arroyo Guevara, A. M., Mcquillan, D., Richter, R. B., Salazar, G., &#38; Sullivan, M. (2021). Drawings of complete graphs in the projective plane. <i>Journal of Graph Theory</i>. Wiley. <a href=\"https://doi.org/10.1002/jgt.22665\">https://doi.org/10.1002/jgt.22665</a>","ieee":"A. M. Arroyo Guevara, D. Mcquillan, R. B. Richter, G. Salazar, and M. Sullivan, “Drawings of complete graphs in the projective plane,” <i>Journal of Graph Theory</i>, vol. 97, no. 3. Wiley, pp. 426–440, 2021.","chicago":"Arroyo Guevara, Alan M, Dan Mcquillan, R. Bruce Richter, Gelasio Salazar, and Matthew Sullivan. “Drawings of Complete Graphs in the Projective Plane.” <i>Journal of Graph Theory</i>. Wiley, 2021. <a href=\"https://doi.org/10.1002/jgt.22665\">https://doi.org/10.1002/jgt.22665</a>."},"publication_identifier":{"issn":["0364-9024"],"eissn":["1097-0118"]},"status":"public","oa":1,"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.","issue":"3","article_type":"original","external_id":{"arxiv":["2002.02287"],"isi":["000631693200001"]},"_id":"9295","year":"2021","volume":97,"page":"426-440","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","type":"journal_article","scopus_import":"1","quality_controlled":"1","date_published":"2021-03-23T00:00:00Z","author":[{"full_name":"Arroyo Guevara, Alan M","id":"3207FDC6-F248-11E8-B48F-1D18A9856A87","first_name":"Alan M","orcid":"0000-0003-2401-8670","last_name":"Arroyo Guevara"},{"last_name":"Mcquillan","first_name":"Dan","full_name":"Mcquillan, Dan"},{"full_name":"Richter, R. Bruce","first_name":"R. Bruce","last_name":"Richter"},{"full_name":"Salazar, Gelasio","first_name":"Gelasio","last_name":"Salazar"},{"full_name":"Sullivan, Matthew","first_name":"Matthew","last_name":"Sullivan"}],"article_processing_charge":"No","doi":"10.1002/jgt.22665","date_updated":"2023-08-07T14:26:15Z","isi":1,"month":"03","language":[{"iso":"eng"}],"project":[{"call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"day":"23","main_file_link":[{"url":"https://arxiv.org/abs/2002.02287","open_access":"1"}],"intvolume":"        97","publication_status":"published"}]