Removing Independently Even Crossings Michael Pelsmajer IIT Chicago

Removing Independently Even Crossings Michael Pelsmajer IIT Chicago Marcus Schaefer De. Paul University Daniel Štefankovič University of Rochester

Crossing number cr(G) = minimum number of crossings in a drawing* of G cr(K 5)=1 *(general position drawings, i. e. , no intersections with 3 edges, edges don’t cross vertices, edges do not touch)

Crossing number poorly understood, for example: ● don’t know cr(Kn), cr(Km, n) Guy’s conjecture: cr(Kn)= Zarankiewicz’s conjecture: cr(Km, n)= ● no approximation algorithm

Pair crossing number pcr(G) = minimum number of pairs of edges that cross in a drawing* of G pcr(K 5)=1 *(general position drawings, i. e. , no intersections with 3 edges, edges don’t cross vertices, edges do not touch)

Odd crossing number ocr(G) = minimum number of pairs of edges that cross oddly in a drawing* of G ocr(K 5)=1 oddly = odd number of times *(general position drawings, i. e. , no intersections with 3 edges, edges don’t cross vertices, edges do not touch)

Rectilinear crossing number rcr(G) = minimum number of crossings in a planar straight-line drawing of G rcr(K 5)=1

“Independent” crossing numbers only non-adjacent edges contribute iocr(G)=minimum number of pairs of non-adjacent edges that cross oddly in a drawing of G ocr(G) = minimum number of pairs of edges that cross oddly in a drawing of G

“Independent” crossing numbers only non-adjacent edges contribute iocr(G)=minimum number of pairs of non-adjacent edges that cross oddly in a drawing of G What should be the ordering of edges around v? v “independent’’ does not matter!

iocr(G)=CVP {e 0, e 1} any initial drawing (v, g) 1 0 if g=ei and v is an endpoint of e 1 -i otherwise columns = pair of non-adjacent edges, e. g. , for K 5, 15 columns rows = non-adjacent (vertex, edge), e. g. , for K 5, 30 rows

iocr(G)=CVP {e 0, e 1} any initial drawing (v, g) 1 0 if g=ei and v is an endpoint of e 1 -i otherwise columns = pair of non-adjacent edges, e. g. , for K 5, 15 columns rows = non-adjacent (vertex, edge), e. g. , for K 5, 30 rows

Crossing numbers iocr(G) acr(G) pcr(G) ocr 0 1 0 0 acr 0 1 0 2 cr(G) rcr(G) pcr 0 1 1 1 cr 0 1 2 2

Crossing numbers iocr(G) acr(G) ocr(G) – amazing fact pcr(G) rcr(G) iocr(G)=0 rcr(G)=0 iocr(G)=0 rcr(G)=0 (Hanani’ 34, Tutte’ 70) (Steinitz, Rademacher’ 34; Wagner ’ 36; Fary’ 48; Stein’ 51)

Crossing numbers iocr(G) acr(G) ocr(G) – amazing fact pcr(G) rcr(G) iocr(G) 2 rcr(G)=iocr(G) 2 cr(G)=iocr(G) (present paper) cr(G) 3 rcr(G)=cr(G) (Bienstock, Dean’ 93)

Crossing numbers - separation iocr(G) acr(G) ocr(G) pcr(G) Tóth’ 08 Pelsmajer, Schaefer, Štefankovič’ 05 different maybe equal? rcr(G) Guy’ 69 cr(K 8) =18, rcr(K 8)=19

Crossing numbers - separation BIG iocr(G) acr(G) ocr(G) pcr(G) very different maybe equal? rcr(G) Bienstock, Dean ’ 93 ( k 4)( G) cr(G)=4, rcr(G)=k

Crossing numbers - separation BIG iocr(G) acr(G) ocr(G) polynomially related very different maybe equal? rcr(G) Bienstock, Dean ’ 93 ( k 4)( G) cr(G)=4, rcr(G)=k Pach, Tóth’ 00 cr(G) (2 ocr(G) ) 2

Crossing numbers - separation BIG iocr(G) acr(G) ocr(G) polynomially related our result very different cr(G) different (2 iocr(G) ) 2 maybe equal? rcr(G) Bienstock, Dean ’ 93 ( k 4)( G) cr(G)=4, rcr(G)=k Pach, Tóth’ 00 cr(G) (2 ocr(G) ) 2

our result very different cr(G) different (2 iocr(G) ) 2 e is bad if f such that ● e, f independent ● e, f cross oddly drawing D realizing iocr(G) bad edges good edges |bad| 2 iocr(G)

drawing D realizing iocr(G) bad edges good edges |bad| 2 iocr(G) GOAL: drawing D’ such that • good edges are intersection free • pair of bad edges intersects 1 times

drawing D realizing iocr(G) bad edges good edges even edges |bad| 2 iocr(G) GOAL: drawing D’ such that • good edges are intersection free • pair of bad edges intersects 1 times

drawing D realizing iocr(G) bad edges Lemma (Pelsmajer, Schaefer, Stefankovic’ 07)good edges even edges cycle C consisting of even edges |bad| 2 iocr(G) redrawing so that C is intersection free, no new odd pairs, same rotation GOAL: drawing D’ such that system • good edges are intersection free • pair of bad edges intersects 1 times

good even, locally bad edges good edges even edges |bad| 2 iocr(G) cycle of good edges cycle of even edges intersection free cycle

good even, locally bad edges good edges even edges |bad| 2 iocr(G) cycle of good edges cycle of even edges intersection free cycle

good even, locally bad edges good edges even edges |bad| 2 iocr(G) cycle of good edges cycle of even edges intersection free cycle

good even, locally bad edges good edges even edges |bad| 2 iocr(G) cycle of good edges cycle of even edges intersection free cycle degree 3 vertices

good even, locally bad edges good edges even edges |bad| 2 iocr(G) cycle of good edges cycle of even edges intersection free cycle degree 3 vertices

good even, locally cycle of good edges cycle of even edges intersection free cycle degree 3 vertices repeat, repeat potentials decreasing: = dv 3 #good cycles with intersections DONE good edges in cycles are intersection free

DONE good edges in cycles are intersection free bad edges good edges not in a good cycle

look at the blue faces bad edges good edges not in a good cycle

add violet good edges, no new faces bad edges good edges not in a good cycle

add bad edges in their faces. . . bad edges good edges not in a good cycle

Open problems Is pcr(G)=cr(G) ? D A B C D A on annulus? B C

Open problems Is iocr(G)=ocr(G) ? (genus g strong Hannani-Tutte) Does iocrg(G)=0 ? Is cr(G)=O(iocr(G)) ?