a crossing lemma for the pair-crossing number
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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
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