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Issue UPDATE: in graph theory, different definitions of edge crossing numbers - impact on applications?


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cr}(G)={\rm pcr}(G)$
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the Electronic Journal of Combinatorics
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Turán’s Brick Factory Problem
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The Graph Crossing Number
The Electronic Journal of Combinatorics

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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


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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%
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SOURCE: https://mathoverflow.net/questions/366765/issue-update-in-graph-theory-different-definitions-of-edge-crossing-numbers
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Summary

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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