---
_id: '9295'
abstract:
- lang: eng
  text: "Hill's Conjecture states that the crossing number  cr(\U0001D43E\U0001D45B)
    \ of the complete graph  \U0001D43E\U0001D45B  in the plane (equivalently, the
    sphere) is  14⌊\U0001D45B2⌋⌊\U0001D45B−12⌋⌊\U0001D45B−22⌋⌊\U0001D45B−32⌋=\U0001D45B4/64+\U0001D442(\U0001D45B3)
    . 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
    \ \U0001D45B4/64+\U0001D442(\U0001D45B3) , thus matching asymptotically the conjectured
    value of  cr(\U0001D43E\U0001D45B) . Let  cr\U0001D443(\U0001D43A)  denote the
    crossing number of a graph  \U0001D43A  in the projective plane. Recently, Elkies
    proved that the expected number of crossings in a naturally defined random projective
    plane drawing of  \U0001D43E\U0001D45B  is  (\U0001D45B4/8\U0001D70B2)+\U0001D442(\U0001D45B3)
    . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked
    if  lim\U0001D45B→∞ cr\U0001D443(\U0001D43E\U0001D45B)/\U0001D45B4=1/8\U0001D70B2
    . We construct drawings of  \U0001D43E\U0001D45B  in the projective plane that
    disprove this."
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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Alan M
  full_name: Arroyo Guevara, Alan M
  id: 3207FDC6-F248-11E8-B48F-1D18A9856A87
  last_name: Arroyo Guevara
  orcid: 0000-0003-2401-8670
- first_name: Dan
  full_name: Mcquillan, Dan
  last_name: Mcquillan
- first_name: R. Bruce
  full_name: Richter, R. Bruce
  last_name: Richter
- first_name: Gelasio
  full_name: Salazar, Gelasio
  last_name: Salazar
- first_name: Matthew
  full_name: Sullivan, Matthew
  last_name: Sullivan
citation:
  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>
  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>.
  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.
  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.
date_created: 2021-03-28T22:01:41Z
date_published: 2021-03-23T00:00:00Z
date_updated: 2023-08-07T14:26:15Z
day: '23'
department:
- _id: UlWa
doi: 10.1002/jgt.22665
ec_funded: 1
external_id:
  arxiv:
  - '2002.02287'
  isi:
  - '000631693200001'
intvolume: '        97'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2002.02287
month: '03'
oa: 1
oa_version: Preprint
page: 426-440
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: Journal of Graph Theory
publication_identifier:
  eissn:
  - 1097-0118
  issn:
  - 0364-9024
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Drawings of complete graphs in the projective plane
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 97
year: '2021'
...