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

2. 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)

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

4. 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)

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

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

7. “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

8. “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? “independent’’  does not matter! v

9. iocr(G)=CVP {e 0,e 1 } (v,g) 1 if g=e i and v is an endpoint of e 1-i 0 otherwise any initial drawing 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

10. iocr(G)=CVP {e 0,e 1 } (v,g) 1 if g=e i and v is an endpoint of e 1-i 0 otherwise any initial drawing 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

11. Crossing numbers ocr(G) acr(G) pcr(G) cr(G) rcr(G) iocr(G)       01000100 01020102 01110111 01220122 ocr acr pcr cr

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

27. good  even, locally cycle of good edges  cycle of even edges  intersection free cycle  degree 3 vertices repeat, repeat, repeat =  d v 3 #good cycles with intersections potentials decreasing: DONE  good edges in cycles are intersection free

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

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

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

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

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

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