colouring maps

Colouring MapsAuthor(s): Richard BridgesSource: Mathematics in School, Vol. 30, No. 3 (May, 2001), pp. 31-34Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212167 .

Accessed: 05/04/2014 20:00

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 65.27.200.210 on Sat, 5 Apr 2014 20:00:31 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=mathas

http://www.jstor.org/stable/30212167?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Colouring Maps by Richard Bridges

This article looks at the relationship between whether or not a map or 'graph' can be drawn without taking the pencil off the paper, and the number of colours needed to colour in its regions. It is strongly suggested that you try out the examples on the accompanying worksheet before reading any further!

Traversability The first aspect of the problem is whether or not we can find a route which traverses every side, or 'edge', of the map in sequence so that no edge is traversed twice. If so the map is said to be traversable. If, as a bonus, it is possible to make the finishing point the same as the start, the map is unicursal. The first mathematician to look into questions like this seriously was Leonhard Euler, and so the two types of traverse are called, respectively, an Euler Tour and an Euler Cycle.

If you have tried the examples on the worksheet you should be able to see when a map is traversable. It is clear that a 'vertex' on the map with three edges leaving it will be a problem. If a prospective tour arrives along one edge and leaves along another, when it comes back again it will have nowhere else to go (Fig. 1). There are exactly two ways to cope with this problem. If the vertex was used as the starting point of the tour the third edge will already have been traversed. Alternatively, the tour can finish here, providing all other edges have been traversed already. The same is true of vertices with any odd number of edges; an even number is traversed on each visit of the tour, leaving a single final edge.

Even vertices create no difficulty; if there was a way in, there must be a way out. The only problem is if we do something silly (Fig. 2), leaving a piece of the map with no way back to it. This can always be corrected, however (Figure 3). One apparent problem is that if we start the tour at an even vertex, an odd number of edges is left behind. But that just means we have to finish there too, forming an Euler Cycle (Fig. 4).

The final result is that a map is unicursal if it has only even vertices, traversable if it has exactly two odd vertices, and is otherwise not traversable. (What about a map with only one odd vertex? Try it! Also, an obvious restriction is that the map must be connected - all in one piece.)

Map Colouring

The edges of our maps divide the plane up into regions (sometimes called 'faces'), and the idea is to colour them in so that no two adjacent regions have the same colour (though the same colour is allowed to meet at a vertex), using as few colours as possible. If you have tried the examples on the worksheet you should have found that all the unicursal maps needed only two colours, the traversable ones needed three, and the non-traversable ones needed three or four. (Map 13

needed five? Try it again!) The main point of this article is to give a proof of the results about unicursal and traversable maps; to justify that they will always be true regardless of how complex we make the maps.

The basic idea is to reduce any map to a simpler and simpler form, but one that at each stage clearly needs the same number of colours. Eventually the map is reduced to such a simple form that the number of colours needed is obvious, yet at the same time we know this is the same number of colours needed for the original map.

The method of reducing a map to a simpler form is shown in Figure 5, where a 4-vertex is removed by merging a pair of non-adjacent regions. It also shows how whatever number of colours is needed for the reduced map will be enough when the original vertex is restored. What about 5-vertices, 6-vertices, etc? Figure 6 shows reduction from a 5-vertex to a 3-vertex, which cannot be further reduced; the two odd vertices in a traversable map will always reduce to two 3- vertices. We have to be careful though; Figure 7 shows the wrong way to reduce a 6-vertex, as it creates two odd vertices where there were none before. Another problem is shown in Figure 8; the pair of regions which has been chosen for merging actually meets elsewhere on the map. Merging them as shown would create a map with a 'self-adjacent region', which is impossible to colour according to our rules. In fact we have to rule out such maps from the start, but we must certainly not create one as we go along. A similar problem is shown in Figure 9, where by choosing to merge two appearances of the same region, part of the map becomes disconnected.

Both these problems are easily avoided as shown in Figure 10, but the key question is: can we always avoid such problems in this way? Fortunately the answer is yes. The reason is that although some pairs of regions meeting at a vertex may also meet elsewhere on the map, or may even be the same region, we can always find a pair of regions which is not like this. For a 'problem' pair of regions will form a closed curve leaving the vertex and coming back to it again (see Figure 10), so that any other pair of regions at the same vertex, of which one is on the inside of this curve and one on the outside, cannot possibly meet each other, and so may be safely merged.

The procedure we must adopt at each vertex, therefore, is to choose a pair of regions, separated by one other region and not meeting elsewhere on the map, and to merge them; this is guaranteed to be possible. The map before merging needs only the same number of colours as the simpler map after merging. The procedure reduces a vertex of degree n to one of n-2 (so a 4-vertex reduces to a 2-vertex, which is just a line), and it remains either even or odd, as it was before. The procedure also keeps the map connected, and does not introduce self-adjacent faces.

Mathematics in School, May 2001 The MA web site www.m-a.org.uk 31



1 r-----a---- 2

ii ,4 " !i ,/..." ,\

"_-..___ ___/

4 II 5

I / ,

,1I ,,;"-,,.

10

Ii |'1 '! p'1 1

II,,

" , .

I

._.------__---,

;..

/I

~)

I ~ I , I h....

ri

a'"\7

/"

'"'x""

%- ,, i

'i-

, | r

., I

Ir

L" .," --

-- , '

i

Ii I

, ,,, .. ...

":

/ i

I II I

i

L .. .::

., , ,..._- ' ,( -. _ _ _ _ _

' " /

" " ....

i

... ." _.---

' ,

-_

-"

"...

___.,_,,...I

3

f

|i

''"" " A

I N,

9i ,II

I

"t /

,I.

,".-

_ _--.:

,...

v

12~

-- ,

I l--..," '',_-

-a '

/

,. ,..

..- ,.. . _ ----- -

a _... .., -,,

.a"'

-. '-

,..- Map Colouring and Traversability For each of the 'maps' on this sheet find out:

(A) Whether you can draw it without taking your pencil off the paper, and without going over any line twice. If you can, and you finish where you started, the map is UNICURSAL. If you can, but you finish in a different place, it is TRAVERSABLE.

(B) The least number of colours needed to colour in each region, including round the outside, so that the same colour never appears on both sides of the same line. The same colour may meet itself at a point.

32 Mathematics in School, May 2001 The MA web site www.m-a.org.uk



Figure 1

/ 7

"i,a 1

2

Figure 2

I

I *

Figure 3

... .. ilL ;

Figure 4

START#' 6 1lEND

..,A r

a

Figure 5

Ttmn staf

. .n

Figure 6 Figure 7

Figure 8 Figure 9

Figure 10 Figure 11

131iC1~D~B~~"Ir

Mathematics in School, May 2001 The MA web site www.m-a.org.uk 33



So what do we end up with if this procedure is repeated as often as possible? Even vertices are eventually removed completely, and odd vertices are reduced to degree three. If the map was originally unicursal, it had only even vertices, so the reduced map has no vertices at all, and yet is connected. The only such map is a simple closed curve, which obviously only needs two colours - one inside and one outside. If the map was originally only traversable, the reduced map has exactly two vertices, both of degree three, is connected, and has no self-adjacent face. The only such map is Example 2 on the worksheet (subject to insignificant distortions - try and find alternatives). We have therefore proved that a unicursal map requires exactly two colours, and a traversable map exactly three.

What of non-traversable maps? This is a problem with an interesting history, of which we will give the briefest sketch. The problem was first raised as a serious mathematical question by F Guthrie in 1852. He proposed the Four Colour Conjecture - that no more than four colours would ever be needed. A proposed proof by A. Kempe in 1879 was shown to be false by P Heawood in 1890, who was, however, able to prove that five colours would always be sufficient. Yet no one had ever found a map that needed more than four colours! For the next 86 years the conjecture remained with this tantalizing gap between four and five, as one of the most famous unsolved problems in mathematics. Mathematicians had almost resigned themselves to thinking it was insoluble when, in 1976, K. Appel and W. Haken finally published a correct proof that four colours always is enough, turning the Four Colour Conjecture into the Four Colour Theorem at last. Yet even their proof was controversial: for the first time in mathematics a computer had played a significant role in checking through over 1000 map configurations!

Where Next?

If you would like to read more about the four colour problem, Keith Devlin's fascinating book (Devlin, 1988) has a chapter all about it. He also shows how to prove the Five Colour Theorem, with little greater difficulty than the proof above.

Map colouring needn't be restricted to the plane. In fact any planar map may be transferred to the surface of a sphere (the region surrounding the map becoming a region 'round the back' of the sphere), and conversely the edges, vertices and faces of polyhedra may be turned into a map (called a Schlegel diagram). Only one of the five Platonic solids has a traversable map - which? Do all the others need four colours? Four of their maps feature amongst the worksheet examples - which? Do any/many unicursal polyhedra exist? What about merely traversable polyhedra? Constructing solid models may help your imagination, or be a satisfying way of exhibiting solutions.

Finally, our result on two-colour maps is the reason why the technique shown in Figure 11 works. Sometimes known as 'op-art', a foreground design consisting of simple closed curves is superimposed on a background also consisting of closed curves. The resulting vertices are all even and so the striking two-colouring is guaranteed to be possible (though the underlying map may be disconnected and hence not traversable). M

Reference

Devlin, K. 1988 Mathematics: the New Golden Age, Pelican.

Keywords: Euler; Traversability; Map Colouring.

Author Richard Bridges, King Edward's School, Birmingham B15 2UA.

REVIEWS REVIEWS REVIEWS The Number Devil Hans Magnus Enzensberger Granta Books 1998 ISBN 1 86207 391 0 260 pages, 240mm x 160mm, softback a12

'If 33 bakers can make 89 pretzels in 21/2 hours, then how many pretzels can 53/4 bakers make in 11/2 hours?' This clearly tongue-in-cheek question reveals two things about The Number Devil, firstly that Enzensberger has a sense of humour and secondly that the translation is American. Do not let the latter fact put you off for this book is not only humorous and amusing but it is fun to read and is surprisingly informative.

Robert does not particularly enjoy mathematics at school nor does he like Mr Bockel, his overweight teacher who eats too many pretzels. But his life is about to change. Robert has twelve dreams in each of which he meets the Number Devil. The horned and be-whiskered Number Devil introduces Robert to a number of mathematical ideas and concepts and causes Robert to become interested in the subject. But 'If you give me homework in my dream, I'll scream bloody murder. That's child abuse,' says Robert, on first meeting the Number Devil.

In Robert's first dream the Number Devil explains infinitely large numbers (1+1+1+1 ...) and infinitely small ones ((1/1+141 ...) and what happens if you take 11 x 11 or 111 x 111. In the second dream there is a brief history lesson on the development of numbers (Roman, Arabic, positive, negative and zero), on powers and place values. During the third night the Number Devil introduces prima donna numbers (primes) and Eratosthene's sieve mentioning Goldbach's conjecture along the way.

In the fourth dream Robert is confronted with unreasonable numbers (irrationals) and recurring and non-recurring decimals. The following night the Number Devil shows Robert triangular numbers, their connections with square numbers and other properties, such as that any number can be made from the sum of two or three triangular numbers.

Rabbits fill Robert's sixth dream as the Number Devil explains Bonacci numbers (Fibonacci). There is an excellent drawing of a tree whose numbers of branches follow Fibonacci. With the knowledge of so many different types of number Pascal's triangle looms in the seventh dream. It is a wonderful source of amusement and Enzensberger exploits it well with normal numbers, triangular ones, powers of 2 and many others. The illustration when only even numbers are highlighted is a powerful image. Permutations and combinations and

their links with Pascal's triangle dominate the eighth dream.

For some the ninth dream could be more of a nightmare. The Number Devil teases Robert with countable infinite sets. Fortunately the next dream is more relaxing with snowflakes, golden ratio, continued fractions and Euler's formula. In the eleventh dream the Number Devil explains something about the ideas of proof and the slow but steady development of mathematics, including a page of Lord Rustle's 1+1=2 (i.e. Russell and Whitehead's Principia Mathematica).

Robert's final dream sees him recognized as a member of the 'Order of Pythagoras, fifth class' and on his way to becoming a mathematician.

So, for whom is this book written? Anybody with a little knowledge of mathematics will find it fun and entertaining. The style is light and easy to read and the ideas develop in a simple but effective way. The illustrations that pepper the book are excellent and in keeping with the witty narrative. It is a book that could be read to Key Stage 3 or 4 classes on Friday afternoons - capturing their attention and introducing and developing mathematical ideas. It is a book that should be in every school's mathematics department.

John Sykes

34 Mathematics in School, May 2001 The MA web site www.m-a.org.uk



colouring maps

Documents