A crossing lemma for the pair-crossing number

1. A crossing lemma for the pair-crossing number Eyal Ackerman and Marcus Schaefer

2. weaker than advertised A crossing lemma for the pair-crossing number Eyal Ackerman and Marcus Schaefer

3. a variant of A crossing lemma for the pair-crossing number Eyal Ackerman and Marcus Schaefer

4. The crossing lemma • The crossing number of a graph , , is the minimum number of edge crossings in a drawing of in the plane. • Crossing Lemma: For every graph with vertices and edges . [Ajtai, Chvátal, Newborn, Szemerédi1982; Leighton 1983] • Tight, up to . Pach& Tóth 97 Pach& Tóth 97 A. 2013 folklore Pach et al. 06

5. The crossing lemma • Proof: • Consider a drawing with crossings • Pick every vertex with probability and get • Plug in the expected values and set • The crossing number of a graph , , is the minimum number of edge crossings in a drawing of in the plane. • Crossing Lemma: For every graph with vertices and edges . [Ajtai, Chvátal, Newborn, Szemerédi1982; Leighton 1983] • Tight, up to . Pach& Tóth 97 Pach& Tóth 97 A. 2013 folklore Pach et al. 06

6. Other crossing numbers • – min number of crossings when is drawn with straight-line edges. • – min number of pairs of edges that cross. • – min number of pairs of edges that cross oddly. • And many more… [Schaefer 2013]

7. Adjacent crossings • Are adjacent crossings redundant? • Tutte: “… crossings of adjacent edges are trivial, and easily got rid of”. • True for but notnecessarily for other variants. • Pach and Tóth: • Rule +: Adjacent crossings are not allowed. • Rule -: Adjacent crossings are not counted. • Rule 0: Adjacent crossings are allowed and counted. • Fulek et al. , Adjacent crossings do matter, GD2011: there are graphs such that -.

8. Other crossing lemmas -+ Using the probabilistic proof and the strong Hanani-Tutte Theorem Thm: .* Thm: +.* * If is not too sparse.

9. Improving via local crossing number • The local crossing number of a graph , , is the minimum such that can be drawn with at most crossings per edge. • Or: = minimum such that is -planar. • Improving the crossing lemma: • Prove that if is “small” then is “sparse”. • E.g., if then . • Use it to get a “weak” bound . • E.g., • Use the weak bound instead of in the probabilistic proof of the crossing lemma.

10. Improving via local crossing number (2) • [Euler] [Pach & Tóth 1997] [Pachet al. 2006] [A. 2013]

11. The local pair-crossing number • The local pair-crossing number of a graph , , is the minimum such that can be drawn with every edge crossing at most other edges (each of them possibly more than once). • Clearly, . • It can happen that :

12. vs. • [Schaefer & Štefankovič 2004] • Thm: If then . • Cor: • Just saw: . • Open: ? • If true, then implies . • Thm: if then .

13. Improving the crossing lemma for pcr+ • Using the bounds on the size of graphs with small we get: + • Plugging this bound into the probabilistic proof yields + for .

14. Proving • since • – a drawing of with the least number of crossings such that . • Suppose that is crossed 3 times: • No consecutive crossings with the same edge:

15. Proving • since • – a drawing of with the least number of crossings such that . • Suppose that is crossed 3 times: • Crossing pattern must be :

16. Summary and open problems • A pair-crossing lemma: For every graph with vertices and edges + • Does it hold for ? • Is it true that +? • Is it true that ? • Known: [Matousek 2013]

17. Summary and open problems (2) • Is it true that ? • Thm: If then . • There is such that . • Open: ? • Thm: if then . • What about the local odd-crossing number? • ?

18. Thank you and