---
_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
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. Journal of Graph Theory. 2021;97(3):426-440.
doi: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
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.
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.
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.” Journal of Graph Theory, vol. 97, no. 3, Wiley, 2021, pp. 426–40,
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.
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'
...