Drawings of complete graphs in the projective plane
Hill's Conjecture states that the crossing number cr ( K n ) of the complete graph K n in the plane (equivalently, the sphere) is 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ = n 4 ∕ 64 + O ( n 3 ). Moon proved that the expected number of crossings in a spherical drawing in which the points...
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Published in | Journal of graph theory Vol. 97; no. 3; pp. 426 - 440 |
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Main Authors | , , , , |
Format | Journal Article |
Language | English |
Published |
Hoboken
Wiley Subscription Services, Inc
01.07.2021
|
Subjects | |
Online Access | Get full text |
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Summary: | Hill's Conjecture states that the crossing number
cr
(
K
n
) of the complete graph
K
n in the plane (equivalently, the sphere) is
1
4
⌊
n
2
⌋
⌊
n
−
1
2
⌋
⌊
n
−
2
2
⌋
⌊
n
−
3
2
⌋
=
n
4
∕
64
+
O
(
n
3
). 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
n
4
∕
64
+
O
(
n
3
), thus matching asymptotically the conjectured value of
cr
(
K
n
). Let
cr
P
(
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
K
n is
(
n
4
∕
8
π
2
)
+
O
(
n
3
). In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if
lim
n
→
∞
cr
P
(
K
n
)
∕
n
4
=
1
∕
8
π
2. We construct drawings of
K
n in the projective plane that disprove this. |
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ISSN: | 0364-9024 1097-0118 |
DOI: | 10.1002/jgt.22665 |