# The four colour theorem

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The Four Colour Conjecture first seems to have been made by Francis Guthrie. He was a student at University College London where he studied under De Morgan. After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan. Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them.

De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:-

A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented. ...... If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did....

Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):-

I am not likely to attempt your quaternion of colour very soon.

Before continuing with the history of the Four Colour Conjecture we will complete details of Francis Guthrie. After practising as a barrister he went to South Africa in 1861 as a Professor of Mathematics. He published a few mathematical papers and became interested in botany. A heather (Erica Guthriei) is named after him.

De Morgan kept asking if anyone could find a solution to Guthrie's problem and several mathematicians worked on it. Charles Peirce in the USA attempted to prove the Conjecture in the 1860's and he was to retain a lifelong interest in the problem. Cayley also learnt of the problem from De Morgan and on 13 June 1878 he posed a question to the London Mathematical Society asking if the Four Colour Conjecture had been solved. Shortly afterwards Cayley sent a paper *On the colouring of maps* to the Royal Geographical Society and in was published in 1879. The paper explains where the difficulties lie in attempting to prove the Conjecture.

On 17 July 1879 Alfred Bray Kempe announced in *Nature* that he had a proof of the Four Colour Conjecture. Kempe was a London barrister who had studied mathematics under Cayley at Cambridge and devoted some of his time to mathematics throughout his life. At Cayley's suggestion Kempe submitted the Theorem to the *American Journal of Mathematics* where it was published in 1879. Story read the paper before publication and made some simplifications. Story reported the proof to the Scientific Association of Johns Hopkins University in November 1879 and Charles Peirce, who was at the November meeting, spoke at the December meeting of the Association of his own work on the Four Colour Conjecture.

Kempe used an argument known as *the method of Kempe chains*. If we have a map in which every region is coloured red, green, blue or yellow except one, say *X*. If this final region *X* is not surrounded by regions of all four colours there is a colour left for *X*. Hence suppose that regions of all four colours surround *X*. If *X* is surrounded by regions *A*, *B*, *C*, *D* in order, coloured red, yellow, green and blue then there are two cases to consider.

(i) There is no chain of adjacent regions from

AtoCalternately coloured red and green.

(ii) There is a chain of adjacent regions fromAtoCalternately coloured red and green.

If (i) holds there is no problem. Change *A* to green, and then interchange the colour of the red/green regions in the chain joining *A*. Since *C* is not in the chain it remains green and there is now no red region adjacent to *X*. Colour *X* red.

If (ii) holds then there can be no chain of yellow/blue adjacent regions from *B* to *D*. [It could not cross the chain of red/green regions.] Hence property (i) holds for *B* and *D* and we change colours as above.

Kempe received great acclaim for his proof. He was elected a Fellow of the Royal Society and served as its treasurer for many years. He was knighted in 1912. He published two improved versions of his proof, the second in 1880 aroused the interest of P G Tait, the Professor of Natural Philosophy at Edinburgh. Tait addressed the Royal Society of Edinburgh on the subject and published two papers on the (what we should now call) Four Colour Theorem. They contain some clever ideas and a number of basic errors.

The Four Colour Theorem returned to being the Four Colour Conjecture in 1890. Percy John Heawood, a lecturer at Durham England, published a paper called *Map colouring theorem.* In it he states that his aim is

rather destructive than constructive, for it will be shown that there is a defect in the now apparently recognised proof.

Although Heawood showed that Kempe's proof was wrong he did prove that every map can be 5-coloured in this paper. Kempe reported the error to the London Mathematical Society himself and said he could not correct the mistake in his proof. In 1896 de la Vallée Poussin also pointed out the error in Kempe's paper, apparently unaware of Heawood's work.

Heawood was to work throughout his life on map colouring, work which spanned nearly 60 years. He successfully investigated the number of colours needed for maps on other surfaces and gave what is known as the *Heawood estimate* for the necessary number in terms of the *Euler characteristic* of the surface.

Heawood's other claim to fame is raising money to restore Durham Castle as Secretary of the Durham Castle Restoration Fund. For his perseverance in raising the money to save the Castle from sliding down the hill on which it stands Heawood received the O.B.E.

Heawood was to make further contributions to the Four Colour Conjecture. In 1898 he proved that if the number of edges around each region is divisible by 3 then the regions are 4-colourable. He then wrote many papers generalising this result.

To understand the later work we need to define some concepts.

Clearly a graph can be constructed from any map the regions being represented by the vertices and two vertices being joined by an edge if the regions corresponding to the vertices are adjacent. The resulting graph is planar, that is can be drawn in the plane without any edges crossing. The Four Colour Conjecture now asks if the vertices of the graph can be coloured with 4 colours so that no two adjacent vertices are the same colour.

From the graph a triangulation can be obtained by adding edges to divide any non-triangular face into triangles. A configuration is part of a triangulation contained within a circuit. An unavoidable set is a set of configurations with the property that any triangulation must contain one of the configurations in the set. A configuration is reducible if it cannot be contained in a triangulation of the smallest graph which cannot be 4-coloured.

The search for avoidable sets began in 1904 with work of Weinicke. Renewed interest in the USA was due to Veblen who published a paper in 1912 on the Four Colour Conjecture generalising Heawood's work. Further work by G D Birkhoff introduced the concept of reducibility (defined above) on which most later work rested.

Franklin in 1922 published further examples of unavoidable sets and used Birkhoff's idea of reducibility to prove, among other results, that any map with ≤ 25 regions can be 4-coloured. The number of regions which resulted in a 4-colourable map was slowly increased. Reynolds increased it to 27 in 1926, Winn to 35 in 1940, Ore and Stemple to 39 in 1970 and Mayer to 95 in 1976.

However the final ideas necessary for the solution of the Four Colour Conjecture had been introduced before these last two results. Heesch in 1969 introduced the *method of discharging*. This consists of assigning to a vertex of degree *i* the charge 6 - *i*. Now from Euler's formula we can deduce that the sum of the charges over all the vertices must be 12. A given set *S* of configurations can be proved unavoidable if for a triangulation *T* which does not contain a configuration in *S* we can redistribute the charges (without changing the total charge) so that no vertex ends up with a positive charge.

Heesch thought that the Four Colour Conjecture could be solved by considering a set of around 8900 configurations. There were difficulties with his approach since some of his configurations had a boundary of up to 18 edges and could not be tested for reducibility. The tests for reducibility used Kempe chain arguments but some configurations had obstacles to prevent reduction.

The year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time. The proof was achieved by Appel and Haken, basing their methods on reducibility using Kempe chains. They carried through the ideas of Heesch and eventually they constructed an unavoidable set with around 1500 configurations. They managed to keep the boundary ring size down to ≤ 14 making computations easier that for the Heesch case. There was a long period where they essentially used trial and error together with unbelievable intuition to modify their unavoidable set and their discharging procedure. Appel and Haken used 1200 hours of computer time to work through the details of the final proof. Koch assisted Appel and Haken with the computer calculations.

The Four Colour Theorem was the first major theorem to be proved using a computer, having a proof that could not be verified directly by other mathematicians. Despite some worries about this initially, independent verification soon convinced everyone that the Four Colour Theorem had finally been proved. Details of the proof appeared in two articles in 1977. Recent work has led to improvements in the algorithm.

**References (9 books/articles)**

**Other Web sites:**

- New Brunswick, Canada

- Georgia, USA (A new simpler proof)

- A S Calude

- J Wilson (A Gresham lecture - also available as a Video version)

**Article by:** *J J O'Connor* and *E F Robertson*