Crossing numbers: connections and some open problems

1. Crossing numbers: connections and some open problems Laszlo A. Szekely University of South Carolina Supported in part by NSF DMS 071111

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

3. Turan’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. Turan remembering 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-Toth (1998): “Which crossing number is it anyway?” (Mohar 1995) • ImreLakatos: “Proofs and Refutations” applied Popper to mathematics analysing Euler’s polyhedral formula and the concept of real function

6. 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)?

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

8. Turan 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)

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

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

11. Extremal graph theory • Pach, Spencer, Toth (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.

12. Bisection Width Method vs. Crossing Lemma • Pach, Spencer, Toth (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

13. Embedding (G2)

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

15. Crossing number method • Spencer-Szemeredi-Trotter (1984) n points in the plane determine at most O(n4/3)unit distances. • Proof: 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.

16. Pictures of G and G2 G G2

17. Unit distances • Erdos-Hickerson-Pach (1989) • Valtr • Embedding approach: consider the distance 2multigraphG2 from G above. For simplicity, assume d unit circles passes through every point.

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

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

21. Bisection width • Leighton (1982), Sykora-Vrto (1993), Pach-Shahrokhi-Szegedy (1994)

22. Embedding • Leighton (1982) • Shahrokhi-Sykora-S-Vrto (1994) • Assume H is embedded into G by ω. • Assume G is drawn.

23. Convex crossing numbers • Convex (outerplanar, 1-page) crossing number: Place vertices on a circle and join them with straight line segments • Shahrokhi, Sykora, S, Vrto (2004), Czabarka, Sykora, S, Vrto (2006) Assume f(x) is edge isoperimetric function for G and Δf is non-negative and decreasing till middle. Then

24. Algorithmic issues • cr(G) ≤ k NP-complete Garey-Johnson (1983) • Pair crossing number ≤ k NP-hard Pach-Toth (1998) • Odd crossing number ≤ k NP-complete Pach-Toth (1998) • Leighton-Rao (1988) For degree bounded graphs, log4n times approximation algorithm for |cr(G) − n| • Even-Guha-Schieber (2000) For degree bounded graphs, log3n times approximation algorithm for |cr(G) − n|

25. Algorithmic issues • cr(G) ≤ k for fixed kin quadratic time: Grohe (2004) • In linear time: Kawabarayashi and Reed (2007)

26. Incidence bipartite graph G • Two vertex sets: n points and m straight lines • Edges: incidences • Szemeredi-Trotter (1983) • |E(G)|=O(n+m+(nm)2/3) • Theorem: de Caen, S (1997) • 3-path PcP’c’ # P3=O(nm) • Implies Szemeredi-Trotter through Atkinson-Watterson-Moran (1960)

27. Incidence bipartite graph v’ v v’ P P’ v P P’

28. A conjecture • Is # C6=O(nm) in the incidence bipartite graph of points and straight lines? • This would imply # P3=O(nm)

29. Bipartite crossing number and linear arrangement • G bipartite graph • Place vertex classes on two parallel lines • Draw edges in straight line segments • Bipartite crossing number bcr(G) • Linear arrangement value L(G) sum of stretches • Shahrokhi-Sykora-S-Vrto (2001) under proper assumptions

30. Circular arrangement problem • For the convex (outerplanar, page-1) crossing number • Czabarka, Sykora, S, Vrto: minimize • Similar lower bounds for this as for convex crossing number, changing Δ2f to Δ2F • Shahrokhi’s problem: does cr(G) have something to do with linear arrangement?