Odd Crossing Number

1
Odd Crossing Number is NOT Crossing Number
Michael Pelsmajer IIT (Chicago) Marcus
Schaefer DePaul University (Chicago) Daniel
Štefankovic University of Rochester
2
Crossing number
cr(G) minimum number of crossings
in a planar drawing of G
cr(K5)?
3
Crossing number
cr(G) minimum number of crossings
in a planar drawing of G
cr(K5)1
4
Rectilinear crossing number
rcr(G) minimum number of crossings
in a planar straight-line drawing
of G
rcr(K5)?
5
Rectilinear crossing number
rcr(G) minimum number of crossings
in a planar straight-line drawing
of G
rcr(K5)1
6
Rectilinear crossing number
rcr(G) minimum number of crossings
in a planar straight-line drawing
of G
cr(G) ? rcr(G)
7
cr(G)0 ? rcr(G)0
THEOREM SR34,W36,F48,S51
Every planar graph has a straight-line planar
drawing.
Steinitz, Rademacher 1934 Wagner 1936 Fary
1948 Stein 1951
8
Are they equal?
cr(G)0 , rcr(G)0
cr(G)1 , rcr(G)1
cr(G)2 , rcr(G)2
cr(G)3 , rcr(G)3
?
cr(G)rcr(G)
9
cr(G) ? rcr(G)
THEOREM Guy 69
cr(K8) 18 rcr(K8)19
cr(G)rcr(G)
10
cr(G) ? rcr(G)
THEOREM Guy 69
cr(K8) 18 rcr(K8)19
THEOREM Bienstock,Dean 93
(8k)(9G) cr(G) 4
rcr(G)k
11
Crossing numbers
cr(G) minimum number of crossings
in a planar drawing of G
rcr(G) minimum number of crossings
in a planar straight-line drawing
of G
cr(G) ? rcr(G)
(?G) cr(G) ? rcr(G)
12
Odd crossing number
ocr(G) minimum number of pairs of edges
crossing odd number of times
13
Odd crossing number
ocr(G) minimum number of pairs of edges
crossing odd number of times
ocr(G) ? cr(G)
14
Odd crossing number
ocr(G) minimum number of pairs of edges
crossing odd number of times
ocr(K5)?
15
Proof (Tutte70) ocr(K5)1
INVARIANT
How many pairs of non-adjacent edges intersect
(mod 2) ?
16
Proof (Tutte70) ocr(K5)1
17
Proof ocr(K5)1
How many pairs of non-adjacent idges intersect
(mod 2) ?
steps which change isotopy
18
Proof ocr(K5)1
How many pairs of non-adjacent idges intersect
(mod 2) ?
steps which change isotopy
19
Proof ocr(K5)1
How many pairs of non-adjacent idges intersect
(mod 2) ?
20
Proof ocr(K5)1
How many pairs of non-adjacent idges intersect
(mod 2) ?
QED
21
Hanani34,Tutte70
ocr(G)0 ? cr(G)0
If G has drawing in which all pairs of edges
cross even times ? graph is planar!
22
Are they equal?
ocr(G)0 , cr(G)0
QUESTION Pach-Tóth00
?
ocr(G)cr(G)
23
Are they equal?
ocr(G)0 ? cr(G)0
?
ocr(G)cr(G)
Pach-Tóth00
cr(G) ? 2ocr(G)2
24
Main result
THEOREM Pelsmajer,Schaefer,Š 05
ocr(G) ? cr(G)
25
How to prove it?
THEOREM Pelsmajer,Schaefer,Š 05
ocr(G) ? cr(G)

1. Find G.
2. Draw G to witness small ocr(G).
3. Prove cr(G)gtocr(G).

26
How to prove it?
THEOREM Pelsmajer,Schaefer,Š 05
ocr(G) ? cr(G)

1. Find G.
2. Draw G to witness small ocr(G).
3. Prove cr(G)gtocr(G).

Obstacle cr(G)gtx is co-NP-hard!
27
Crossing numbers for maps
28
Crossing numbers for maps
29
Crossing numbers for maps
30
Ways to connect
31
Ways to connect
32
Ways to connect
33
Ways to connect
34
Ways to connect
35
Ways to connect
number of Dehn twists
-1
0
1
36
Ways to connect
How to compute intersections ?
37
Ways to connect
How to compute intersections ?
0
2
1
38
Crossing number
min ?iltjxi-xj(?igt?j)
xi2Z
do arcs i,j intersect in the initial drawing?
the number of twists of arc i
39
Crossing number
i
min ?iltjxi-xj(?igt?j)
xi2Z
j
do arcs i,j intersect in the initial drawing?
the number of twists of arc i
40
Crossing number
min ?iltjxi-xj(?igt?j)
xi2Z
j
i
do arcs i,j intersect in the initial drawing?
the number of twists of arc i
41
Crossing number
min ?iltjxi-xj(?igt?j)
OPT
xi2Z
OPT
xi2R
42
Crossing number
min ?iltjxi-xj(?igt?j)
OPT
xi2Z
OPT
xi2R
Lemma OPT OPT.
43
Crossing number
min ?iltjxi-xj(?igt?j)
Lemma OPT OPT.
Obstacle cr(G)gtx is co-NP-hard!
44
Crossing number
min ?iltjxi-xj(?igt?j)
yij xi-xj(?igt?j)
yij xixj-(?igt?j)
Obstacle cr(G)gtx is co-NP-hard!
45
Crossing number
min ?iltj yij
yij xi-xj(?igt?j)
yij xixj-(?igt?j)
Obstacle cr(G)gtx is co-NP-hard!
46
Crossing number
Dual linear program
max ?iltj Qij(?igt?j)
QT-Q Q10 -1 ? Qij ? 1
Q is an nn matrix
47
EXAMPLE
a
b
c
d
48
Odd crossing number ?
a
b
c
d
49
Odd crossing number
a
ocr ? adbc
b
c
d
50
Crossing number ?
a
max ?iltj Qij(?igt?j)
b
QT-Q Q10 -1 ? Qij ? 1
c
d
?(2,1,4,3)
a ? b ? c ? d ac ? d
0 ac b(d-a) -ac 0
ab a(c-b) b(a-d) -ab 0 bd
a(b-c) -bd 0
cr ? acbd
51
Putting it together
a
ocr ? adbc
b
cr ? acbd
c
a ? b ? c ? d ac ? d
d
bc1, a(v3-1)/20.37, dac
ocr/crv3/20.87
52
Crossing number
a
ocr/crv3/20.87
b
c
d
53
Crossing number
a
ocr/crv3/20.87
b
c
for graphs?
d
54
Crossing number
a
ocr/crv3/20.87
b
c
d
cr?
55
Crossing number
a
ocr/crv3/20.86
b
c
d
cr?
56
Crossing number for graphs
Theorem (8 ?gt0) (9 graph) with
ocr/cr ? v3/2?.
57
Is crO(ocr)?
58
Is crO(ocr)?
Is cr O(ocr) on annulus?
59
Is crO(ocr)?
Is cr O(ocr) on annulus?
Theorem On annulus cr ? 3ocr
60
Theorem On annulus cr ? 3ocr
BAD triple
GOOD triple
61
n.CR ? 3BAD
p
BAD triple
Pay of bad triples p,i,j
Average over p.
62
BAD ? n.OCR
random i,j,k Xodd pairs
BAD triple
BAD bin(n,3)
3OCR bin(n,2)
EX
?
?
63
BAD ? n.OCR
n.CR ? 3BAD
BAD triple
CR ? 3OCR
(on annulus)
64
Crossing number for graphs
There exists graph with
ocr/cr ? v3/2?.
On annulus ocr/cr ? 1/3
Experimental evidence ocr/cr ? v3/2 on
annulus and pair of pants
Bold (wrong) conjecture For any graph
ocr/cr ? v3/2
65
Questions
crossing number of maps with d vertices in
poly-time? (true for d ? 2)
Bold (wrong) conjecture For any graph
ocr/cr ? v3/2
(map graph rotation system)
66
Open questions - classic
Guys conjecture
cr(Kn)
Zarankiewiczs conjecture
cr(Km,n)
Better approx algorithm for cr.
67
Crossing number for graphs
pair crossing number (pcr) number of pairs of
crossing edges algebraic crossing number (acr)
? algebraic crossing number of edges
-1
1
68
Crossing numbers
acr(G)
ocr(G)
cr(G)
rcr(G)
pcr(G)