{"publication_identifier":{"eissn":["1097-0118"],"issn":["0364-9024"]},"year":"2021","abstract":[{"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.","lang":"eng"}],"publisher":"Wiley","issue":"3","ec_funded":1,"status":"public","article_type":"original","department":[{"_id":"UlWa"}],"oa_version":"Preprint","type":"journal_article","citation":{"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.","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.","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","short":"A.M. Arroyo Guevara, D. Mcquillan, R.B. Richter, G. Salazar, M. Sullivan, Journal of Graph Theory 97 (2021) 426–440.","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.","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."},"date_created":"2021-03-28T22:01:41Z","scopus_import":"1","isi":1,"article_processing_charge":"No","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.","title":"Drawings of complete graphs in the projective plane","month":"03","date_updated":"2023-08-07T14:26:15Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2002.02287"}],"oa":1,"quality_controlled":"1","language":[{"iso":"eng"}],"intvolume":" 97","page":"426-440","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publication":"Journal of Graph Theory","publication_status":"published","day":"23","date_published":"2021-03-23T00:00:00Z","project":[{"name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"754411"}],"external_id":{"isi":["000631693200001"],"arxiv":["2002.02287"]},"volume":97,"_id":"9295","doi":"10.1002/jgt.22665","author":[{"first_name":"Alan M","full_name":"Arroyo Guevara, Alan M","last_name":"Arroyo Guevara","orcid":"0000-0003-2401-8670","id":"3207FDC6-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Mcquillan, Dan","last_name":"Mcquillan","first_name":"Dan"},{"first_name":"R. Bruce","full_name":"Richter, R. Bruce","last_name":"Richter"},{"first_name":"Gelasio","last_name":"Salazar","full_name":"Salazar, Gelasio"},{"first_name":"Matthew","last_name":"Sullivan","full_name":"Sullivan, Matthew"}]}