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TSE  March 2001

TSE March 2001

Subject:

Re: (VERY) OFF TOPIC - Map coloring

From:

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Date:

Mon, 12 Mar 2001 09:35:22 EST

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Pat Sloane wrote about the four-color map problem (3/12/2001):

> What's the book? The reason the proof was controversial is that
> it isn't actually a "proof" in any logical or mathematical sense. 
> Just a brute force examining of all examples the computer could
> think of, and no proof (or way of proving) that the program was
> actually sufficient to test "every" possible example. 
>
> Personally, I think the map-coloring problem is a design problem,
> not a mathematical problem

Not a mathematical problem? You have absolutely no idea what you are talking 
about. 

Professor Appel is one of the two researchers who proved the four-color map 
theorem using computers. I was a grad student at the University of Illinois 
in 1976, taking a graduate math course from Professor Appel, when his proof 
was publicly announced.

For those interested in this VERY off-topic topic, here's some info on the 
problem, with the work of Professor Appel cited towards the end of the 
article.:

-- Steve --

From:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem

..html

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 A to C alternately 
coloured red and green.
(ii) There is a chain of adjacent regions from A to C alternately 
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. 

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

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