1. Odd Crossing Number is NOT Crossing Number Michael Pelsmajer IIT (Chicago) Marcus Schaefer DePaul University (Chicago) Daniel Štefankovič University of Rochester

2. Crossing number cr(G) = minimum number of crossings in a planar drawing of G cr(K 5 )=?

3. Crossing number cr(G) = minimum number of crossings in a planar drawing of G cr(K 5 )=1

4. Rectilinear crossing number rcr(G) = minimum number of crossings in a planar straight-line drawing of G rcr(K 5 )=?

5. Rectilinear crossing number rcr(G) = minimum number of crossings in a planar straight-line drawing of G rcr(K 5 )=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 Every planar graph has a straight-line planar drawing. Steinitz, Rademacher 1934 Wagner 1936 Fary 1948 Stein 1951 THEOREM [SR34,W36,F48,S51]:

8. cr(G)=0, rcr(G)=0 cr(G)=rcr(G) ? cr(G)=1, rcr(G)=1 cr(G)=2, rcr(G)=2 cr(G)=3, rcr(G)=3 Are they equal?

9. cr(G)  rcr(G) cr(K 8 ) =18 rcr(K 8 )=19 THEOREM [Guy’ 69]: cr(G)=rcr(G)

10. cr(G)  rcr(G) cr(K 8 ) =18 rcr(K 8 )=19 THEOREM [Guy’ 69]: THEOREM [Bienstock,Dean ‘93]: ( 8 k)( 9 G) cr(G) =4 rcr(G)=k

11. 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 Crossing numbers (G) cr(G)  rcr(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(K 5 )=?

15. Proof (Tutte’70): ocr(K 5 )=1 How many pairs of non-adjacent edges intersect (mod 2) ? INVARIANT:

16. Proof (Tutte’70): ocr(K 5 )=1

17. Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ? steps which change isotopy:

18. Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ? steps which change isotopy:

19. Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ?

20. Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ? QED

21. Hanani’34,Tutte’70: ocr(G)=0  cr(G)=0 If G has drawing in which all pairs of edges cross even # times  graph is planar!

22. ocr(G)=0, cr(G)=0 ocr(G)=cr(G) ? Are they equal? QUESTION [Pach-Tóth’00]:

23. ocr(G)=0  cr(G)=0 ocr(G)=cr(G) ? Are they equal? Pach-Tóth’00: cr(G)  2ocr(G) 2

24. THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G)  cr(G) Main result

25. THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G)  cr(G) 1. Find G. 2. Draw G to witness small ocr(G). 3. Prove cr(G)>ocr(G). How to prove it?

26. THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G)  cr(G) Obstacle: cr(G)>x is co-NP-hard! 1. Find G. 2. Draw G to witness small ocr(G). 3. Prove cr(G)>ocr(G). How to prove it?

27. Crossing numbers for “maps”

30. Ways to connect

35. 0+1 Ways to connect number of “Dehn twists”

36. Ways to connect How to compute # intersections ?

37. 0 1 2 Ways to connect How to compute # intersections ?

38. Crossing number min  i  j )| xi2Zxi2Z the number of twists of arc i do arcs i,j intersect in the initial drawing?

39. Crossing number min  i  j )| xi2Zxi2Z the number of twists of arc i do arcs i,j intersect in the initial drawing? i j

40. Crossing number min  i  j )| xi2Zxi2Z the number of twists of arc i do arcs i,j intersect in the initial drawing? i j

41. Crossing number min  i  j )| xi2Zxi2Z xi2Rxi2R OPT OPT *

42. Crossing number min  i  j )| xi2Zxi2Z xi2Rxi2R Lemma: OPT * = OPT. OPT OPT *

43. Crossing number min  i  j )| Lemma: OPT * = OPT. Obstacle: cr(G)>x is co-NP-hard!

44. Crossing number min  i  j )| Obstacle: cr(G)>x is co-NP-hard! y ij ¸ x i -x j +( i > j ) y ij ¸ –x i +x j -( i > j )

45. Crossing number min  i<j y ij Obstacle: cr(G)>x is co-NP-hard! y ij ¸ x i -x j +( i > j ) y ij ¸ –x i +x j -( i > j )

46. Crossing number max  i  j ) Q T =-Q Q1=0 -1 Q ij  1 Dual linear program Q is an n £ n matrix

47. EXAMPLE: a c b d

48. Odd crossing number ? a c b d

49. Odd crossing number a c b d ocr  ad+bc

50. Crossing number ? max  i  j ) a c b d a  b  c  d a+c  d (2,1,4,3) cr  ac+bd 0 ac b(d-a) * -ac 0 ab a(c-b) b(a-d) -ab 0 bd * a(b-c) -bd 0 Q T =-Q Q1=0 -1 Q ij  1

51. Putting it together a c b d ocr  ad+bc cr  ac+bd a  b  c  d a+c  d b=c=1, a=(√3-1)/2~0.37, d=a+c ocr/cr=√3/2~0.87

52. Crossing number a c b d ocr/cr=√3/2~0.87

53. Crossing number a c b d ocr/cr=√3/2~0.87 for graphs?

54. Crossing number a c b d ocr/cr=√3/2~0.87 cr=?

55. Crossing number a c b d ocr/cr=√3/2~0.86 cr=?

56. Crossing number for graphs ocr/cr  √3/2+. Theorem: ( 8 >0) ( 9 graph) with

57. Is cr=O(ocr)?

58. Is cr  O(ocr) on annulus?

59. Is cr=O(ocr)? Is cr  O(ocr) on annulus? Theorem: On annulus cr  3ocr

60. BAD tripleGOOD triple Theorem: On annulus cr  3ocr

61. BAD triple n.CR  3#BAD p Pay: #of bad triples {p,i,j} Average over p.

62. BAD triple #BAD  n.OCR random i,j,k X=#odd pairs E[X] #BAD bin(n,3) 3OCR bin(n,2) 

63. BAD triple n.CR  3#BAD #BAD  n.OCR CR  3OCR (on annulus)

64. Crossing number for graphs ocr/cr  √3/2+. There exists graph with On annulus ocr/cr  1/3 Experimental evidence: ocr/cr  √3/2 on annulus and pair of pants Bold (wrong) conjecture: For any graph ocr/cr  √3/2

65. Questions crossing number of maps with d vertices in poly-time ? (true for d  2) (map = graph + rotation system) Bold (wrong) conjecture: For any graph ocr/cr  √3/2

66. Open questions - classic Zarankiewicz’s conjecture: cr(K m,n ) Guy’s conjecture: cr(K 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

68. Crossing numbers ocr(G) acr(G) pcr(G) cr(G) rcr(G)