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