If you're reading a proof that seems too good to be true, you should quickly get a sense of the whole argument, then focus on the parts that seem likely to be wrong.
You shouldn't take a portion of the proof that's correct, and optimize it. But that's what I've just done.
This new claimed proof of the 4-color theorem, just 6 pages long, uses a formula for counting rooted planar maps. But there's a simpler formula that would greatly simplify some of the arguments!
(1/n)
A 'planar map' is a graph embedded in the sphere. This is what the 4-color theorem is about. We can always draw such a map on the plane, but we should think of it on the sphere. Then one of the faces - that is, countries - extends out to infinity.
A planar map become 'rooted' when we choose one of its edges and draw an arrow on it. We can always draw the map so this special edge is touching the infinite face, as shown here.
How many rooted planar maps with n edges are there?
(2/n)
In 1963 the famous combinatorist Tutte showed that there are
πβ = 2 3βΏ (2π)! / π!(π+2)!
rooted planar maps with n edges. Jackson and Richmond's claimed proof of the 4-color theorem uses a formula for the generating function
π΄(π₯) = ββ πβπ₯βΏ
Namely, they say it equals the hypergeometric function βπΉβ[Β½, 1 ; 3; 12π₯] .
I think that's right. (They subtract 1, but we can worry about that later.)
But there seems to be a much simpler formula for π΄(π₯), which makes everything easier!
(3/n)
If Jackson and Richmond had looked up the sequence πβ on the On-Line Encyclopedia of Integer Sequences:
they'd have seen Simon Plouffe gave a simpler formula for its generating function in 2012:
\( \displaystyle{ A(x) = \frac{-1 + 18x - (1 - 12x)^{3/2}}{54 x^2} } \)
This simpler formula would make some of their arguments a lot easier!
(4/n)
For example, Jackson and Richmond use the following complicated argument involving the hypergeometric function βπΉβ to show π΄(π₯) has the form shown in equation (7).
But this fact is instantly obvious once we know
\( \displaystyle{ A(x) = \frac{-1 + 18x - (1 -12x)^{3/2}}{54 x^2} } \)
(5/n)
By the way, I probably got a bunch of small stuff wrong.
For example, should we count the rooted planar tree with zero edges, which gives the constant term in the generating function π΄(π§)? It seems Jackson and Richmond don't, which is why they subtract 1 from βπΉβ[Β½, 1 ; 3; 12π₯] . But the On-Line Encyclopedia of Integer Sequences does.
This could be straightened out.
(6/n)
By now it's clear this paper is wrong in several ways, and cannot be salvaged. To learn why, start here:
https://mathstodon.xyz/@noamzoam/109567981846531700
and read the comments by Noam Zeilberger (@noamzoam) and Simon Pepin Lehalleur (@plain_simon). Then I carefully describe a whole set of *other* errors.
But anyway, it was fun learning about the generating function for rooted planar maps.
By the way, the pictures of rooted planar maps came from this cool paper:
https://arxiv.org/abs/1408.5028
(7/n, n = 7)
@johncarlosbaez @highergeometer I've been reading the paper, and a basic element of this strategy seems problematic to me. Namely, that the proportion of planar maps that are 4-colorable *is* exponentially small! A planar map has a proper 4-coloring of its faces (or dually, vertices) iff it is bridgeless (or dually, loopless). Now, the #s of (rooted) planar maps and of bridgeless/loopless planar maps were counted by Tutte, corresponding to https://oeis.org/A000168 and https://oeis.org/A000260.
@SvenGeier - part (3/n) of my tweets has no LaTeX, just Unicode. I do that for people like you.
The stuff with \displaystyle in part (4/n) is LaTeX. That equation would have been hard to write in Unicode, since I'm raising something to the 3/2 power. Unicode lets you write π₯Β³Β², but I don't think they've got a superscript /.
Guilherme Duarte
John Carlos Baez
Noam Zeilberger
Simon Pepin Lehalleur
Ξ£(iΒ³) = (Ξ£i)Β²
Ianagol