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Odd Crossing Number is NOT Crossing Number Michael Pelsmajer IIT (Chicago) Marcus Schaefer DePaul University (Chicago) Daniel Štefankovič University of Rochester
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Crossing number cr(G) = minimum number of crossings in a planar drawing of G cr(K 5 )=?
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Crossing number cr(G) = minimum number of crossings in a planar drawing of G cr(K 5 )=1
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Rectilinear crossing number rcr(G) = minimum number of crossings in a planar straight-line drawing of G rcr(K 5 )=?
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Rectilinear crossing number rcr(G) = minimum number of crossings in a planar straight-line drawing of G rcr(K 5 )=1
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Rectilinear crossing number rcr(G) = minimum number of crossings in a planar straight-line drawing of G cr(G) rcr(G)
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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]:
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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?
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cr(G) rcr(G) cr(K 8 ) =18 rcr(K 8 )=19 THEOREM [Guy’ 69]: cr(G)=rcr(G)
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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
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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)
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Odd crossing number ocr(G) = minimum number of pairs of edges crossing odd number of times
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Odd crossing number ocr(G) = minimum number of pairs of edges crossing odd number of times ocr(G) cr(G)
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Odd crossing number ocr(G) = minimum number of pairs of edges crossing odd number of times ocr(K 5 )=?
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Proof (Tutte’70): ocr(K 5 )=1 How many pairs of non-adjacent edges intersect (mod 2) ? INVARIANT:
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Proof (Tutte’70): ocr(K 5 )=1
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Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ? steps which change isotopy:
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Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ? steps which change isotopy:
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Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ?
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Proof: ocr(K 5 )=1 How many pairs of non-adjacent idges intersect (mod 2) ? QED
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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!
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ocr(G)=0, cr(G)=0 ocr(G)=cr(G) ? Are they equal? QUESTION [Pach-Tóth’00]:
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ocr(G)=0 cr(G)=0 ocr(G)=cr(G) ? Are they equal? Pach-Tóth’00: cr(G) 2ocr(G) 2
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THEOREM [Pelsmajer,Schaefer,Š ’05] ocr(G) cr(G) Main result
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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?
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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?
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Crossing numbers for “maps”
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Ways to connect
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0+1 Ways to connect number of “Dehn twists”
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Ways to connect How to compute # intersections ?
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0 1 2 Ways to connect How to compute # intersections ?
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Crossing number min i j )| xi2Zxi2Z the number of twists of arc i do arcs i,j intersect in the initial drawing?
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Crossing number min i j )| xi2Zxi2Z the number of twists of arc i do arcs i,j intersect in the initial drawing? i j
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Crossing number min i j )| xi2Zxi2Z the number of twists of arc i do arcs i,j intersect in the initial drawing? i j
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Crossing number min i j )| xi2Zxi2Z xi2Rxi2R OPT OPT *
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Crossing number min i j )| xi2Zxi2Z xi2Rxi2R Lemma: OPT * = OPT. OPT OPT *
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Crossing number min i j )| Lemma: OPT * = OPT. Obstacle: cr(G)>x is co-NP-hard!
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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 )
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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 )
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Crossing number max i j ) Q T =-Q Q1=0 -1 Q ij 1 Dual linear program Q is an n £ n matrix
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EXAMPLE: a c b d
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Odd crossing number ? a c b d
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Odd crossing number a c b d ocr ad+bc
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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
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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
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Crossing number a c b d ocr/cr=√3/2~0.87
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Crossing number a c b d ocr/cr=√3/2~0.87 for graphs?
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Crossing number a c b d ocr/cr=√3/2~0.87 cr=?
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Crossing number a c b d ocr/cr=√3/2~0.86 cr=?
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Crossing number for graphs ocr/cr √3/2+. Theorem: ( 8 >0) ( 9 graph) with
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Is cr=O(ocr)?
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Is cr O(ocr) on annulus?
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Is cr=O(ocr)? Is cr O(ocr) on annulus? Theorem: On annulus cr 3ocr
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BAD tripleGOOD triple Theorem: On annulus cr 3ocr
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BAD triple n.CR 3#BAD p Pay: #of bad triples {p,i,j} Average over p.
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BAD triple #BAD n.OCR random i,j,k X=#odd pairs E[X] #BAD bin(n,3) 3OCR bin(n,2)
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BAD triple n.CR 3#BAD #BAD n.OCR CR 3OCR (on annulus)
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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
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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
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Open questions - classic Zarankiewicz’s conjecture: cr(K m,n ) Guy’s conjecture: cr(K n ) Better approx algorithm for cr.
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Crossing number for graphs pair crossing number (pcr) # number of pairs of crossing edges algebraic crossing number (acr) algebraic crossing number of edges +1
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Crossing numbers ocr(G) acr(G) pcr(G) cr(G) rcr(G)
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