Claim of Proof to Four-Color Theorem

By G. Spencer-Brown

To the Editors of Nature, 4 Little Essex Street, London W C 2 England
copies to other interested journals

From G Spencer-Brown, 2 St Peters St, Castle Hill, Cambridge England
17 December 1976

Sirs

The recent announcement by two American computer scientists that they have a proof of the four colour theorem, although they certainly have not published a proof, coupled with the fact that they are widely reported as saying they believe that no simple or elegant proof of this theorem is possible, prompts me to refer to the work of me and my brother, the late D J Spencer-Brown, on this theorem as early as 1960-1964.

As reported in 1969 [1], we found during this period an extremely elegant way of expressing the four-colour conjecture (as it then was) which, if verified, would lead to a correspondingly elegant proof.

As is well known, the difficulty of the foul colour problem stems from the fact that the Heawood formulae [2], say Hmin, Hmax, giving the minimum and maximum values for the chromatic numbers of surfaces (Sg) of connectivity g, give

Hmin = Hmax = [(1/2)(7 + (24g - 23)^(1/2) )] for g > 1 ,

but for g = 1 (the spherical surface and the plane) give Hmin = 4 and Hmax = 6 . Heawood was able to show [2] by other means that the maximum number of colours needed in this case was not in fact greater than 5, and no mathematician has since been able to improve on this, although we have all suspected Hmin = 4 to be in this case the correct answer, otherwise the sphere and the plane would be exceptions to the Heawood rule.

Since g - 1 is the only case where the Heawood formulae are no help, my brother and I decided to approach the problem from the reverse direction. Our aim was to give the map in S1 a form that does not allow it more than four colours, and then to prove (if we could) that such a form will generate all maps.

Unfortunately we could get no support for this work, and had to abandon it at the end of 1964. At this stage we had little doubt that the four colour conjecture was true, ard various mathematicians with whom we corresponded, including Bertrand Russell, were aware that we had a method that was almost certainly capable of proving it.

Since the Americans have not published a proof, their announcement of course does not preempt the issue, and leaves the position in respect of the first published proof exactly as before.

It was because of this that a number of my colleagues in the Department of Pure Mathematics in the University of Cambridge persuaded me to have another look at the work I did with my brother twelve years ago to see if, with some further effort, it could be made to yield a proof. I am delighted to report that I have been able to find such a proof and that it will be presented and published in the form of a recorded lecture under the auspices of the University of London on Monday 20th December 1976 and later, of course, written up for presentation in printed form.

The nub of the proof lies in two theorems, the first essential to the method, and the second a crucial restatement of the fourcolour conjecture in a way that renders it susceptable to proof.

Theorem 1, The theorem of primary colourability
If a map is colourable with n colours, it is colourable with p primary colours, p being the least integer such that 2^p >= n.

This is readily seen to be true by imagining the map as made up of primary bounded areas which may overlap to form the distinguishable regions. We now have 2^p possibilities of colouration of the individual regions, ranging from no colouration to colouration with all p primary colours. From this it can be shown that any standard map M is expressible as a structure of simple closed curves in Sg , and taking C(p) as the greatest number of such curves in Sg needed to bound p primary areas in M we now have

for g > 1, Hmax = Hmin, C(p) > p;
for g = 1, Hmax > Xmin, C(p) = p.

The truth of the latter condition of course follows from the Jordan curve theorem, true just in case g = 1.

Thus in case g = 1, p = 2 , although we have lost the Heawood equality, we have gained the Spencer-Brown equality, and this is what renders it possible to prove

Theorem 2. The four colour theorem
Every standard map in a surface of unit connectivity can be expressed as a real construction of two sets of simple closed curves.

This elegant theorem I have now proved, and the proof will have been published by the time you are able to print this communication.

G Spencer-Brown


Notes

  1. G Spencer-Brown, Laws of Form, London 1969, New York 1972, pp xxi. (New York edition), 99.
  2. P J Heawood, "Map-colour theorem," Quart J Math Oxford Ser 24 (1890) 322 - 338