Removing Independently Even Crossings

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(K5)=1 *(general position drawings, i.e., no intersections with 3 edges, edges don’t cross vertices, edges do not touch)

3. 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

4. Pair crossing number pcr(G) = minimum number of pairs of edges that cross in a drawing* of G pcr(K5)=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(K5)=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)

6. Rectilinear crossing number rcr(G) = minimum number of crossings in a planarstraight-line drawing of G rcr(K5)=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. {e0,e1} iocr(G)=CVP any initial drawing (v,g) 1 if g=ei and v is an endpoint of e1-i 0 otherwise columns = pair of non-adjacent edges, e.g., for K5, 15 columns rows = non-adjacent (vertex,edge), e.g., for K5, 30 rows

10. {e0,e1} iocr(G)=CVP any initial drawing (v,g) 1 if g=ei and v is an endpoint of e1-i 0 otherwise columns = pair of non-adjacent edges, e.g., for K5, 15 columns rows = non-adjacent (vertex,edge), e.g., for K5, 30 rows

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

12. Crossing numbers – amazing fact iocr(G) acr(G)    cr(G) rcr(G) ocr(G)    pcr(G) iocr(G)=0  rcr(G)=0 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) acr(G)    cr(G) rcr(G) ocr(G)    pcr(G) iocr(G)  2  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 iocr(G) acr(G) cr(G) rcr(G) ocr(G) pcr(G) Guy’69 cr(K8) =18, rcr(K8)=19 Tóth’08 Pelsmajer, Schaefer, Štefankovič’05 different maybe equal?

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

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

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

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

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

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

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

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

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

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

26. goodeven, locally bad edges good edges even edges |bad|2iocr(G) 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 potentials decreasing: =  dv3 #good cycles with intersections DONE  good edges in cycles are intersection free

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

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

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

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

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

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