Crossing numbers: history, applications to discrete geometry and open problems

1. Crossing numbers: history, applications to discrete geometry and open problems László A. Székely University of South Carolina Supported in part by NSF DMS 071111 GraDR 2012 Crossing Number Workshop and Minischool Valtice, Czech Republic, May 21, 2012

2. Measures of non-planarity • Thickness • Skewness • Splitting number • Vertex deletion number • Page number • Genus • Crossing number

3. Turán’s Brick Factory Problem • “There were some kilns where the bricks were made and some open storage yards where the bricks were stored. All the kilns were connected by rail with all storage yards. … the trouble was only at crossings. The trucks generally jumped the rails there and the bricks fell out of them; in short this caused a lot of trouble and loss of time … The idea occurred to me that this loss of time could be minimized if the number of crossings of the rails had been minimized. But what is the minimum number of crossings?” • (P. Turánremembering in 1977)

4. Zarankiewicz conjecture

5. “Rise and Fall of the Zarankiewicz Theorem” (R. Guy 1969) • Proofs: Zarankiewicz (1954) Urbanik (1955) • Gap found: Ringel and Kainen independently • Pach-Tóth(1998): “Which crossing number is it anyway?” (Mohar 1995) • ImreLakatos: “Proofs and Refutations” applied Popper to mathematics analysingEuler’s polyhedral formula and the concept of real function

6. Guy conjecture • Achieved in two distinct ways: • Soup can drawing • Throwing different slopes to different hemispheres

7. From topological graph theory to asymptotic results (Leighton 1982) • Crossing Lemma • Bisection width lower bound • Graph embedding

8. Crossing Lemma – exploiting Euler’s formula • Crossing Lemma (Leighton, Ajtai-Chvátal-Newborn-Szemerédi): For a simple graph G on n vertices and m edges, either m ≤ 4n or • For a graph G on n vertices and m edges, with edge multiplicity up to M, either m ≤ CMnor

9. Midrange crossing constant Pach, Spencer, Tóth (1999) • If n<< m << n2 converges to a constant (conjectured by Erdős and Guy)

10. Bisection width • Leighton (1982), Sýkora-Vrťo(1993), Pach-Shahrokhi-Szegedy (1994)

11. Graph embedding • G1=(V1,E1), G2=(V2,E2), |V1|≤|V2| • Embedding ω: G1−>G2: a pair of injections (φ,ψ), where φ:V1−>V2and ψ:E1−>(path set of G2) such that ψ(uv) is a φ(u)−φ(v) path for all uv in E1 • Fore εE2, μω(e)=|{f εE1 : e εψ(f)}| • Foru εV2, mω(u)=|{f εE1 : u εψ(f)}| • μω=maxe(μω(e)) μω(e1)=μω(e2)=2 φ G2 G1 mω(u2) =3 μω=2 ψ mω(u1)=mω(u3)=2

12. Embedding induced drawing G2 with drawing D(G2) • Two types of crossings occur in the induced drawing of G1: • At crossing edges of G2 • At vertices of G2 G1 ω Induced drawing ID(G1)

13. Embedding • Leighton (1982) • Shahrokhi-Sýkora-Sz-Vrťo (1994) • Assume G1 is embedded into G2 by ω. • Assume G2 is drawn as D(G2), inducing ID(G1).

14. Turán numbers • Let T(n,k,l,s)denote the minimum size of an l-uniform hypergraph on n vertices, such that any k-element subset contains at least s edges from the hypergraph. • T(n,k,l)= T(n,k,l,1) • Ringel observed that T(n,5,4) ≤ cr(Kn) • Analogously if s ≤ crg(Kp) then T(n,p,4,s) ≤ crg(Kn)

15. Turán’s graph theorem • If G has no k independent nodes T(n,k,2)≤ e(G) • If , the complement, has no k-clique, then

16. Analogy of Turán numbers and crossing numbers • Counting method: count crossings in copies of Kn in a drawn copy of Kn+1: • And hence

17. Analogy of Turán numbers and crossing numbers • Katona-Nemetz-Simonovits(1964) on Turán numbers • Improvement on a fixed-size problem induces infinitely many improvements • deKlerk, Maharry, Pasechnik, Salazar, Richter (2004) 83% of Zarankiewicz conjecture

18. Extremal graph theory • Pach, Spencer, Tóth (1999) improvement on the Crossing Lemma (conjectured by Simonovits): if G has girth >2rand m>4n, then • As m2>cr(G), G has at most edges tight for r=2,3,5.

19. Bisection Width Method vs. Crossing Lemma • Pach, Spencer, Tóth(1999) used the bisection width method to prove their girth theorem • Alternative proof through Crossing Lemma and graph embedding (yielding explicit constant): • Assume that G is drawn in its optimal drawing (and for simplicity assume that G is d-regular) • Define Grby joining vertices of G if their distance is r – they are joined by a unique r-path

20. Embedding (G2)

21. Proof • Draw Grfollowing closely the paths in the drawing of G first second category category

22. Erdős-Purdy Conjecture (1970) • The number of incidences between n points and m lines on the plane is at most • This is tight up to a constant multiplicative factor Number of incidences = 7

23. Szemerédi-Trotter Theorem (1982) • TheErdős-Purdy Conjecture is true. • The number of incidences between n points and m lines on the plane is at most • Alternatively, if n½ ≤ m ≤ n2, then the number of incidences is at most

24. Crossing number method (Sz 1997) • Consider an “appropriate” “natural” graph drawn in the plane. • Crossings, number of vertices or number of edges of this graph should be related to the quantity of interest. • Set lower and upper bound for the crossing number of this graph • Analyze the results in terms of the quantity of interest.

25. Proof of Szemerédi-Trotter Thm • Wlog every line has at least one point. The graph G is already drawn in the plane: • Vertices are the points • Edges are the line segments connecting two consecutive points on any one of the lines • #of incidences on a line = #of edges on that line + 1 • |E(G)| = #of incidences− m

26. Erdős Unit Distance Conj. (1946) • For all ε> 0, there is a constant cε such that the number of unit distances among n points on the plane is at most cεn1+ε • Erdős’ construction shows that the number of unit distances can be n1+(c/loglogn)(which is bigger than nlogn, even bigger than nlogkn). n= 7, 11 unit distances

27. Unit distance upper bounds • cn3/2Erdős 1946 • o(n3/2) Józsa, Szemerédi 1973 • n1.444… Beck, Spencer 1984 • cn4/3 Spencer, Szemerédi, Trotter 1984

28. Spencer-Szemerédi-Trotter (1984) • n points in the plane determine at most O(n4/3)unit distances. • Proof by crossing number method: Draw unit circles around the points and define a drawn graph G. Vertices= the n points, edges connect consecutive points on the circles. m=2# unit distances.

29. A related problem • Among n points and n unit circles in the plane, what is the maximum number of incidences? • The maximum number of incidences has the same magnitude of growth as the maximum number of unit distances among n points in the plane, as n goes to infinity. n= 2, 3 incidences

30. Unit distances • Erdős-Hickerson-Pach(1989) • Valtr • Just topology cannot prove the Erdősunit distance conjecture.

31. Embedding G2 into G G G2

32. Unit distances • Embedding approach: consider the distance 2multigraphG2 from G above. For simplicity, assume d unit circles passes through every point.

33. “Something else” is needed • M<danyway, but M<<d would give improvement • “Something else” is needed • What could it be? • E.g. Elekes-Simonovits-Szabó(2007): • 3 points in the plane, n-n-n unit circles pass through them. # of points covered by all 3 families =O(n2−ε).

35. Further applications of crossing number method/Szemerédi-Trotter • Elekes: n real numbers have at least n5/4 distinct sums or products • Pach, Sharir, and others: many more incidence bounds (points, translates of convex closed curves, etc.) • Andrews (Iosevits proof): For a convex polygon in the plane with n lattice vertices n=O(area1/3) • Dey: # of planar k-sets is at most 7n(k+2)1/3

36. Distinct Distances (1946) • Erdősconjecture (1946): The number of distinct distances among n points on the plane is at least • Erdős’ matching construction is a square grid of size n½×n½.

37. Distinct distance lower bounds • Cn1/2Erdős 1946 • Cn2/3Moser 1952 • Cn5/7Fan Chung 1984 • Cn(58/81)−εBeck 1984 • n4/5/logcnFan Chung, Szemerédi, Trotter 1992 • Cn3/4from a single point Clarkson, Edelsbrunner, Guibas, Sharir, Welzl 1990 • Cn4/5 from a single point Sz 1997 • Cn6/7 Solymosi, Tóth2001 • Cn(4e/(5e-1))-o(1)Tardos 2002 • Cn(19/22)-o(1) Katz 2003 • Cn(48-14e)/(55-16e)-o(1) Katz, Tardos2004 • Cn/lognNets, Katz 2010

38. Natural graph for distinct distances Draw concentric circles around each point with radiuses = distances from point Very high edge-multiplicity is possible:

39. Erdős-Szemerédiproblem • Determine the quantity • Clearly, g(n)=O(n2) • g(n)=Ω(n1+ε) Erdős-Szemerédi1983 • g(n)=Ω(n32/31) Nathanson • g(n)=Ω(n16/15) Ford • g(n)=Ω(n5/4) Elekes 1997

40. Erdős-Szemerédiproblem • Proof by Elekes:

41. Complex Szemerédi-Trotter? • Complex plane:

42. Planar graphs have O(n) edges • Is there a proof not using Euler’s formula that planar graphs drawn in straight line segments have O(n) edges? • Yes – Pinchasi(2007) • Drawing triplet systems in C2: • Every triplet is contained by a complex line • Body of a triplet: convex hull in R4 • Bodies may share vertex or boundary edge • How many triplets can be drawn? O(n)?

43. Embedding abstract simplicial complexes • Abstract simplicial complex: finite family of finite sets, closed for taking subsets • Its dimension: max set size minus 1 • 1-dimensional abstract simplicial complex can be identified with a graph • Embedding in Euclidean space: (d+1)-sets span a d–dimensional simplex; simplices only overlap in spans of common subsets • Embedded 1-dimensional simplicial complex: rectilinear drawing of a graph • Theorem: every d–dimensional abstract simplicial complex embeds in R2d+1, but some does not embed in R2d.

44. Incidence bipartite graph G • Two vertex sets: n points and m straight lines • Edges: incidences • Szemerédi-Trotter (1982) • |E(G)|=O(n+m+(nm)2/3) • Theorem: de Caen, S (1997) • 3-path PcP’c’ # P3=O(nm) • Implies Szemerédi-Trotter through Atkinson-Watterson-Moran (1960)

45. Incidence bipartite graph v’ v v’ P P’ v P P’

46. A “conjecture” • Is # C6=O(nm) in the incidence bipartite graph of points and straight lines? • This would imply # P3=O(nm) • Slightly fails (by loglog nfactor for m=n) Klavík, Kráľ, Mach (2011)

47. Ubiquitous Sum of Degree Squares • Bisection width • Embedding • Convex crossing numbers • Approximation algorithms

48. Bisection width • Leighton (1982), Sýkora-Vrťo(1993), Pach-Shahrokhi-Szegedy (1994)

49. Embedding • Leighton (1982) • Shahrokhi-Sýkora-Sz-Vrťo (1994) • Assume G1 is embedded into G2 by ω. • Assume G2 is drawn as D(G2), inducing ID(G1).