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cr}(G)={\rm pcr}(G)$

Theory Ser

Wikipedia

the Electronic Journal of Combinatorics

Topological Graph Theory

Burlington

Turán’s Brick Factory Problem

The Status of the Conjectures

FK18

S20

The Graph Crossing Number

The Electronic Journal of Combinatorics

Eyal Ackerman

Computational Geometry

RSS

Stack Exchange Inc

Stack Overflow

Ph.D. Thesis

Pach

Tóth

János Pach

Géza Tóth

J. Combin

Schaefer

Hanani–Tutte

Kynčl

L.A. Székely

Hill

R. Gera et al

1.)Hanani–Tutte

Jan Kynčl

Combinatorica

Ramsey

Robertson

Seymour

Kuratowski

Radoslav Fulek

Marcus Schaefer's

Claus Dollinger's

https://dx.doi.org/10.4230/LIPIcs.SoCG.2018.40

J Kynčl

F. Shahrokhi

L. A. Székely

O. Sýkora

I. Vrt'o

Algorithmica 16

László Székely

quoted).I

the Surface of Genus 4

The Graph Crossing Number

https://doi.org/10.37236/2713 - M. Schaefer

Brick Factory Problem

MO

pcr}(G)$

Fulek

Vermont

Kuratowski

the Hanani–

Hanani–Tutte

Positivity 48.00%

Negativity 52.00%

SOURCE:
https://mathoverflow.net/questions/366765/issue-update-in-graph-theory-different-definitions-of-edge-crossing-numbers
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It gives a lower bound for the minimum number of edge crossings ${\rm cr}(G)$ for any drawing of the graph $G$ with $n$ vertices and $e$ edges I double-checked my algorithms and, based on this definition, I clearly apply the pair crossing number ${\rm pcr}(G)$CRITICAL QUESTION: Can you confirm to me that the edge crossing lemma remains valid on the sphere and the torus also for the pair crossing number ${\rm pcr}(G)$?Reference: János Pach and Géza Tóth. Theory Ser. B, 80(2): 225–246, 2000.And Wikipedia article as a starting point https://en.wikipedia.org/wiki/Crossing_number_inequality$\DeclareMathOperator\cr{cr}\DeclareMathOperator\pcr{pcr}$For the pair crossing number $\pcr(G)$, the short answer is yes the crossing lemma holds for drawings on the sphere, but it is not known whether it also holds on the torus.The best and most current reference for you could be the survey article from Schaefer, updated in February 2020: “The Graph Crossing Number and its Variants: A Survey” from the Electronic Journal of Combinatorics (https://doi.org/10.37236/2713).The relevant pages for you are pages 5 and 6 with the following quote from Schaefer:“Since the Hanani–Tutte theorem is not known to be true for the torus, this means that we do not currently have a proof of the crossing lemma for $\pcr$ or $\pcr_−$ on the torus.”Generally, $\pcr(G)\leq \cr(G)$. an asymptotic variant of the crossing lemma, $\operatorname{cr}(G)\ge \Omega(e^3/n^2)$, is true even for the pair crossing number on a fixed surface, such as a torus.With Radoslav Fulek [FK18, Corollary 9] we have shown that [FK18, Claim 5] implies an approximate version of the Hanani–Tutte theorem on orientable surfaces. In particular, [FK18, Claim 5] implies that there is a constant $g$ such that for every graph $G$ that can be drawn on the torus with every pair of independent edges crossing an even number of times, $G$ can be drawn on the orientable surface of genus $g$ without crossings.

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