A crossing lemma for the pair-crossing number

Eyal Ackerman and Marcus Schaefer

A crossing lemma for the pair-crossing number

Eyal Ackerman and Marcus Schaefer

weaker than advertised

A crossing lemma for the pair-crossing number

Eyal Ackerman and Marcus Schaefer

a variant of

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, Szemerédi 1982; Leighton 1983]

Tight, up to .

Pach & Tóth 97A. 2013Pach & Tóth 97 Pach et al. 06folklore

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, Szemerédi 1982; Leighton 1983]

Tight, up to .

Pach & Tóth 97A. 2013Pach & Tóth 97 Pach et al. 06folklore

Proof:

• Consider a drawing with crossings• Pick every vertex with probability and get

• Plug in the expected values and set

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]

Adjacent crossings Are adjacent crossings redundant? Tutte: “… crossings of adjacent edges are trivial, and

easily got rid of”. True for but not

necessarily 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, GD 2011: there are graphs such that - .

Other crossing lemmas

-+ 𝑒3

64𝑛2≤

Using the probabilistic proof and the strong Hanani-Tutte Theorem

Thm: .*

* If is not too sparse.

Thm: +.*

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.

Improving via local crossing number (2)

[Euler]

[Pach & Tóth 1997]

[Pach et al. 2006]

[A. 2013]

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 :

vs.

[Schaefer & Štefankovič 2004]

Thm: If then . Cor:

Just saw: . Open: ?

If true, then implies . Thm: if then .

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 .

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:

Proving

since – a drawing of with the least number of

crossings such that . Suppose that is crossed 3 times:

Crossing pattern must be :

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]

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?

?

Thank you and