@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},
}

@article{5790,
  abstract     = {The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation R′ giving some predrawn chords that represent an induced subgraph of G. The question is whether one can extend R′ to a representation R of the entire graph G, that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of G. Our main result is an O(n3) time algorithm for partial representation extension of circle graphs, where n is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.},
  author       = {Chaplick, Steven and Fulek, Radoslav and Klavík, Pavel},
  issn         = {0364-9024},
  journal      = {Journal of Graph Theory},
  number       = {4},
  pages        = {365--394},
  publisher    = {Wiley},
  title        = {{Extending partial representations of circle graphs}},
  doi          = {10.1002/jgt.22436},
  volume       = {91},
  year         = {2019},
}