TITLE: Issue UPDATE: in graph theory, different definitions of edge crossing numbers - impact on applications?
QUESTION [183 upvotes]: QUICK FINAL UPDATE: Just wanted to thank you MO users for all your support. Special thanks for the fast answers, I've accepted first one, appreciated the clarity it gave me. I've updated my torus algorithm with ${\rm cr}(G)$. Works fine on my full test set, i.e. evidence for ${\rm cr}(G)={\rm pcr}(G)$ on torus. More on this later, will test sharper bound from last answer as well. I'm going to submit in time! Thanks again MO users for all your help!

Original post:
I apologize if „crisis“ is too strong a word, but I am in a mode of panic, if that's the right word: In two weeks, I should be submitting my Ph.D. Thesis, but I have just received bad news, or I should say information that makes me very concerned. It is really an emergency situation:
My thesis is in computer science, algorithms related to graph drawings on the sphere and the torus. One of the cornerstone mathematical results I am relying on is the graph edge crossing lemma (or edge crossing inequality). 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
$${\rm cr}(G)\geq \frac{e^3}{64n^2}$$
for $e>4n$.

PROBLEM: I am reading in the article of Pach and Tóth that there is a possibility that mathematics papers on crossing numbers operate with different definitions. There is the crossing number ${\rm cr}(G)$ (minimum of edge crossings in a drawing of $G$), but also the pair crossing number
${\rm pcr}(G)$, the minimum number of edge pairs crossing in a drawing of $G$.
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. Which crossing number is it anyway? J. Combin. Theory Ser. B, 80(2): 225–246, 2000.
And Wikipedia article as a starting point https://en.wikipedia.org/wiki/Crossing_number_inequality

REPLY [28 votes]: @user161819 I wanted to make a comment but it got too long, so putting it as an answer. But please take it just as a comment for later, once everything is finished:

If I understand your comment to my answer correctly, you are aiming to change your algorithm for the torus so it works with ${\rm cr}(G)$. I think the whole MO community is keeping their fingers crossed, wishing you that you can successfully complete everything in time!

Looking at the far horizon, I wanted to make a suggestion to you. Once you have changed your torus algorithm and completed your thesis, you will have effectively two algorithms in your hands for the torus: The old one based on ${\rm pcr}(G)$ and the new one based on ${\rm cr}(G)$. I am saying the obvious here, keep both of them, they can really be fruitful for future research.
(1) Obviously, your two algorithms could support research on the big open question whether ${\rm pcr}(G)\stackrel{\rm ?}{=}{\rm cr}(G)$ or not. They
could produce experimental evidence, ideas, and insights for a
future proof of equality, or an actual counterexample. (Again, I am
saying the obvious here.)
(2) To really pressure-test ${\rm pcr}(G)\stackrel{\rm ?}{=}{\rm cr}(G)$ on the torus, it would be interesting to also try the
best known to date lower bound for ${\rm cr}(G)$
$$\frac{1}{29}\frac{e^3}{n^2}$$ for graphs with $e>7n$. This lower bound is from
Eyal Ackerman (2019): "On topological graphs with at most four
crossings per edge", Computational Geometry, 85: 101574, 31,
doi:10.1016/j.comgeo.2019.101574 (probably you are aware of it from
the Wikipedia article that you quoted).
I think your question and this whole topic are really important. László Székely calls it one of the "foundational problems" and devotes a whole section to it in his article Turán’s Brick Factory Problem: The Status of the Conjectures of Zarankiewicz and Hill. In: R. Gera et al. (eds.)(2016): Graph Theory—favorite conjectures and open problems. 1.)

For now, fingers crossed that you can complete your thesis in time!