Guy's conjecture, also called the Harary-Hill conjecture, states that the graph crossing number complete graph
(1)
where floor function . Equivalently,
(2)
The values for A028723 ). These are the values
of the conjectured formula ; the actual crossing
numbers are given by OEIS A000241 .
Guy (1972) proved the conjecture for
For odd good drawing
(3)
This parity result is due to Kleitman (1976). Counting each graph crossing in the vertices
gives
(4)
for ceiling function (Pan and Richter 2007).
Consequently, if the conjecture holds for odd
Some optimal good drawings of good
drawings of
OpenAI (2026) announced a proof that formula .
As of Oct. 7, 2026, VibeMathed (2026) had read the formal statement but had
not independently rebuilt it, and independent specialist verification of the complete
proof had not been reported.
Balogh et al. (2026) improved the asymptotic lower
bound to
(5)
Balogh et al. (2026) report ChatGPT assistance with correctness checks in
part of the proof .
See also Complete Bipartite Graph ,
Complete Graph ,
Graph
Crossing Number ,
Zarankiewicz's Conjecture
Explore with Wolfram|Alpha
References Aichholzer, O. "Another Small but Long Step for Crossing Numbers: Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG
2021). pp. 72-77, 2021. https://publications.ist.tugraz.at/abstracts/a-aslscn-21/index.html . Balogh,
J.; Layman, N.; Lidický, B.; Pfender, F.; Salazar, G.; Spiro, S.; and Volec,
J. "Crossing Numbers of Complete Bipartite Graphs." 18 Oct 2026. https://arxiv.org/abs/2610.25204 . Brodsky,
A.; Durocher, S.; and Gethner, E. "The Rectilinear Crossing Number of https://arxiv.org/abs/cs/0009023 . de
Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, R. B.; Salazar, G. "Improved
Bounds for the Crossing Numbers of https://arxiv.org/abs/math/0404142 . de
Klerk, E.; Pasechnik, D. V.; and Schrijver, A. "Reduction of Symmetric
Semidefinite Programs Using the Regular Math Program. 109 , 613-624,
2007. Guy, R. K. "The Crossing Number of the Complete Graph."
Bull. Malayan Math. Soc. 7 , 68-72, 1960. Guy, R. K.
"Crossing Numbers of Graphs." In Graph
Theory and Applications: Proceedings of the Conference at Western Michigan University,
Kalamazoo, Mich., May 10-13, 1972 Harary,
F. and Hill, A. "On the Number of Crossings in a Complete Graph." Proc.
Edinburgh Math. Soc. 13 , 333-338, 1962/1963. Kleitman, D. J.
"A Note on the Parity of the Number of Crossings of a Graph." J. Combin.
Th., Ser. B 21 , 88-89, 1976. https://doi.org/10.1016/0095-8956(76)90032-0 . OpenAI.
"The Crossing Number of Complete Graphs." Sep. 23, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf . Pan,
S. and Richter, R. B. "The Crossing Number of J. Graph Th. 56 , 128-134,
2007. https://doi.org/10.1002/jgt.20249 . Sloane,
N. J. A. Sequences A000241 /M2772
and A028723 in "The On-Line Encyclopedia
of Integer Sequences." VibeMathed. "The Harary-Hill Conjecture
on the Crossing Number of Complete Graphs." Oct. 7, 2026. https://vibemathed.com/problem/harary-hill-conjecture . Referenced
on Wolfram|Alpha Guy's Conjecture
Cite this as:
Weisstein, Eric W. "Guy's Conjecture."
From MathWorld https://mathworld.wolfram.com/GuysConjecture.html
Subject classifications More... Less...
of the complete graph 
satisfies
is the floor function. Equivalently,
,
2, ... are then given by 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441,
588, ... (OEIS A028723). These are the values
of the conjectured formula; the actual crossing
numbers are given by OEIS A000241.
, a result extended to 
by Pan and Richter (2007) and to 
by Aichholzer (2021).
,
any good drawing 
of 
satisfies
subdrawings that retain its four vertices
gives
![cr(K_n)>=[(ncr(K_(n-1)))/(n-4)]](/images/equations/GuysConjecture/NumberedEquation4.svg)
,
where ![[x]](/images/equations/GuysConjecture/Inline13.svg)
is the ceiling function (Pan and Richter 2007).
Consequently, if the conjecture holds for odd 
, it also holds for 
.
cannot be extended to optimal good
drawings of 
(Pan and Richter 2007).
for every 
, attributed to an unreleased internal OpenAI model. The release
supplies a Lean formalization of the complete formula.
As of Oct. 7, 2026, VibeMathed (2026) had read the formal statement but had
not independently rebuilt it, and independent specialist verification of the complete
proof had not been reported.
and 
."
In Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG
2021). pp. 72-77, 2021. 
Is 62." 22 Sep 2000. 
and 
." 2004. 
-Representation." Math Program. 109, 613-624,
2007.Guy, R. K. "The Crossing Number of the Complete Graph."
Bull. Malayan Math. Soc. 7, 68-72, 1960.Guy, R. K.
"Crossing Numbers of Graphs." In 
Is 100." J. Graph Th. 56, 128-134,
2007.