@article{9295,
  abstract     = {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.},
  author       = {Arroyo Guevara, Alan M and Mcquillan, Dan and Richter, R. Bruce and Salazar, Gelasio and Sullivan, Matthew},
  issn         = {1097-0118},
  journal      = {Journal of Graph Theory},
  number       = {3},
  pages        = {426--440},
  publisher    = {Wiley},
  title        = {{Drawings of complete graphs in the projective plane}},
  doi          = {10.1002/jgt.22665},
  volume       = {97},
  year         = {2021},
}