topical paper:wcherowi/courses/m4010/s… · web viewtherefore, a valid question would involve just...

Topical Paper:

The Journey of the Four Color TheoremFirst Draft

by Leah Grant

for partial completion of

MATH 4010

Dr. Cherowitzo

21 April 2005

MATHEMATICIANS AND MAP COLORING ▪ 2

13 Chapter

Mathematiciansand Map Coloring

(1852 – Present)

Introduction

Although an ostensibly simplistic concept, the Four Colour Conjecture has been quite a

difficult one to prove. Escaping both expert mathematicians and amateur enthusiasts, it was not

until more than 100 years after its alleged proposition that an accepted proof was completed. In

the following discussion we will address the individuals surrounding the Four Color Theorem

and their respective contributions to its solution, state the current standing and views on the

problem, and discuss some philosophical implications of its fairly recent computer-aided proofs.

What is the Four Color Theorem?

Maps in general serve to illustrate spatial

orientation of land and water bodies to each other.

Considering their common purposes (showing

direction, suggesting travel routes) they would not

be of much use without clearly differentiating

boundaries between the aforementioned regions.

(2)

MATHEMATICIANS AND MAP COLORING ▪

In attempting this demarcation it is certainly

useful for mapmakers to color adjacent regions

with different colors so as to most discernibly

illustrate them as independent upon initial

inspection. Therefore, a valid question would involve just how many colors are necessary to

produce a map in which adjacent regions are differently colored.

This is the concept at the crux of the FCT, suggesting that all one-dimensional (or planar)

maps, whether existing or imaginary, can be so colored with at most four colors. Oddly enough it

has acquired a large following of mathematicians, rather than mapmakers, as its proof has

progressed. This might be due to lack of encounters with existing land maps of such necessity, or

the fact that there is no truly dire need to color maps with four colors or less. For whatever

reason, the problem has been of only minor interest to those who actually color the items in

question.

That is not to say that mathematicians did not partake in a

great deal of map coloring themselves. It is rumored that FCT

devotee George David Birkhoff spent his entire honeymoon

coloring maps that he insisted his new wife draw for him [7]!

Other enthusiasts developed methods of coloring that

more effectively portrayed colored regions and their boundaries,

such as the edge and vertex method [f]. For the colored “map” (1)

at left, each region A through E can be represented by a node or

“vertex,” and the boundaries between them as “edges.” This way

we can see the orientation of the regions and manipulate their colorings more easily.

1

2

4

Francis Guthrie

Frederick Guthrie

5 ▪ JOURNEY THROUGH GENIUS

Since, as Augustus De Morgan recognized, four regions at most may be in simultaneous

contact with one another, we will need only worry about coloring vertices surrounded by at most

four other vertices [8]. However, the numerous possible arrangements we must consider make

this a very complicated task. So perhaps it is the mathematical complexity of the problem that

suggests it may be better served with attention from the more mathematically inclined.

And this is what it got.

Colorful History

Proofs and contemplation of the FCT were attempted by a host of

mathematicians throughout history. It was Francis Guthrie who first

considered such a concept while coloring maps of England [9]; he later

proposed to his brother Frederick (both students of the famous Augustus De Morgan)…

“..the greatest number of colours to be used in colouring a map so as to

avoid identity of colour in lineally contiguous districts is four…”[7]

It was then Frederick who wrote De Morgan in 1852, asking him for any sort of clarification on

the issue. De Morgan admitted that it

seemed like a plausible assumption,

yet could not figure out a way to

prove it [8]. Slightly puzzled, he

went to his friend Sir William Rowan Hamilton with

the Guthrie brothers’ inquiry…

“…A student of mine asked me to day 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 any how divided and the

Arthur Cayley


compartments differently coloured so that figures with any portion

of common boundary line are differently coloured – four colours

are wanted, but not more… query cannot a necessity for five or

more be invented…” [7]

Yet De Morgan’s letter seemed ineffective at arousing his friend’s curiosity; Sir Hamilton

brusquely replied that he would “…not likely attempt [it] very soon.” [8]

Cayley Comes Calling

De Morgan died in 1871, and interest in the four coloring problem

seemed to fade for a while. Luckily, another of his former students, Arthur

Cayley, had learned of the problem and become very interested in how it

might be proved. In 1878 he addressed the London Mathematical Society

asking whether anyone had supplied a solution to Guthrie’s conjecture [2]. He learned that

Charles Peirce’s had been the first attempt, although he proved quite unsuccessful in his

endeavor [3]. Since no one else seemed to have considered it, responses were less than

satisfactory to Cayley; he felt compelled to explore just what it was about this problem that

eluded even brilliant minds. A while later he published a paper entitled “On the Coloring of

Maps,” explaining his findings on the difficulty of the proof [1].

Kempe’s Attempt

One year after Cayley’s paper was published, and article appeared in the American

Journal of Mathematics. A former student of Cayley’s, Alfred Bray Kempe, seemed to have

found a solution to the four color problem! In “On the geographical problem of the four colours,”

Kempe explained,

6


“… that a very small alteration in one part of a map may render it necessary to

recolour it throughout…after a somewhat arduous search, I have succeeded… in

hitting upon a weak point.” [7]

In his paper, Kempe endeavored to prove by induction that any planar map was four-

colorable. To facilitate this aim, he established some basic but essential conventions; the first of

these employing an extension of a formula by Euler. That is, if we let V represent the vertices of

a region, F denote the region (or face) itself, and E correspond to its number of edges

(boundaries), we have then V + F = E + 1. We can interpret this as saying that the number of

vertices (or “meeting points”) and the number of regions added together are one more than the

number of boundaries in a map.

Using this idea, Kempe showed that (for Rk = the number of regions with k boundaries),

5R1 + 4R2 + 3R3 + 2R4 + R5 - … = 0.

Because only the first five quantities are positive, Kempe argued that R1, R2, R3, R4, and R5 could

not all be zero; we can combine this and Euler’s formula above to deduce that every map must

have a region with fewer than six boundaries (proving de Morgan’s “only five neighbors”

concept).

By this result, we are able to locate a region on any map that is bounded by five or less

other regions. Kempe has us then cover this region with a “patch” of the same (but slightly

larger) shape, and extend the boundaries of the surrounding regions so that they meet at a point

on our patch. We have essentially now reduced that region to a single point, decreasing the

number of countries on our map by one [7].


We repeat this “patching” process until there is just one region left with five or fewer

other regions surrounding it, and we color it any of the four colors. Then we begin stripping our

patches off in reverse order, coloring each re-revealed country with one of our available four

colors. We repeat this process until the original map is restored, now colored with four colors so

that no adjacent regions have the same color.

To ensure this result, Kempe had to consider what would happen when the last remaining

un-patched region was surrounded by different numbers of regions. From before we know that

this region may be bounded by up to five other regions, so we only need to consider five cases.

Suppose we represent our regions with nodes or vertices, their boundaries with lines or edges,

and denote the particular region in question as vertex “v.” When v is bounded only by one other

node, we have two nodes to color (n = 2 case) and can definitely color them with four colors.

When v is surrounded by two or three other nodes (n = 3 and n = 4) four colors still suffice. The

trouble comes when n is surrounded by four or five other nodes (the n = 5 and n = 6 cases), for in

such instances finding a color for v may be tricky.

So how do we know we will encounter these tricky situations? In his proof Kempe

discusses certain arrangements of regions that we are bound to encounter in our coloring

activities, later deemed “unavoidable sets.” Because any map that might jeopardize the four color

theorem must at least contain a region with five or fewer surrounding regions (“only five

neighbors” theorem), Kempe reasoned that the map must then contain region v with one of the

following shapes:

8


We know what to do when we get a digon or triangle-shaped v, for there will always be a fourth

color left with which to color it. However, if v is a square or pentagon shape, we run into trouble.

Kempe’s Chains

So to contend with these dilemmas, Kempe applies a rather clever trick. For the first case

that gives us trouble, the “square” v (or n = 5), we assign the four nodes

surrounding v to be colored red, green, yellow, and blue (see below). Since

there is no color left for v, we must somehow change the color(s) of the

surrounding nodes so that they only use three of the four colors without

rendering the rest of the map impossible to four-color.

In order to do this, Kempe tells us to consider the relationships of the

colored nodes to each other. In the case of the oppositely facing red and

yellow nodes here, we could change the yellow node to red (or vice versa)

as long as the two are not connected by an alternating chain of red and

yellow nodes (left). This way, the four nodes surrounding v are colored with

three colors, and there is a color left for v. In the example at left, we may now change v to

yellow; we have successfully four-colored our map.

And if there is an alternating red and yellow chain connecting

the original red and yellow nodes? Recall that our lines or

edges represent the boundaries between two countries. Therefore,

the lines cannot cross each other, as this would imply that

one country can cross over into another (which would defeat the purpose of

illustrating boundaries in the first place). Thus having such a chain

connecting our red and yellow nodes would prevent the existence of a blue-

green alternating chain connecting the blue and green nodes, as it would have to cross the red-

Percy Heawood


yellow connective string. We could then change the green (or blue) node allowing v to be

colored, thereby successfully four coloring the map (at left). Kempe uses a similar argument in

proving the n = 6 case, where v is a pentagon surrounded by five other nodes colored with all

four colors.

This technique later became known as Kempe’s Chain method, and afforded him great

acclaim in the mathematical world. After the proof appeared in the American Journal of

Mathematics in 1879, Kempe was made a Fellow of the Royal Society and later knighted for his

significant contribution [4].

Heawood’s Hay day

Kempe’s celebrated proof, however, contained a small error. Eleven years

after it had been widely accepted as true, Percy John Heawood (another map

coloring enthusiast) discovered a counterexample map for which Kempe’s chain

method failed. In his famous paper, “Map-Colouring

Theorem,” Heawood explains somewhat apologetically that his “…

aims are so far rather destructive than constructive, for it will be

shown that there is a defect in the now apparently recognized proof…”

[7] He went on to discuss the flaw in Kempe’s logic concerning the

use of his chain method in the pentagonal n = 6 case, using the

example at right (where r = red, g = green, b = blue, y = yellow, and v

is the middle uncolored node).

You will notice that this particular coloring of the 25-region “map” results in the failure

of Kempe’s chain method. The map actually is four colorable if it is colored differently!

Heawood used this arrangement simply to expose the logical error in Kempe’s coloring

technique. There are in fact more simplistic illustrations of such cases, involving fewer regions

10


but where the chain method still fails; most notable are those presented by Alfred Errera and

Charles-Jean-Gustave-Nicolas de la Vallée Poussin, who both discovered the error independent

of Heawood [7].

Let us return again to Heawood, whose intentions in “Map-Colouring Theorem” were not

entirely destructive. Although he was unable to reconstruct a four-color proof from Kempe’s

failed one, he was able to correctly prove that any map on a sphere is five-colorable using

Euler’s formula and a variation on Kempe’s chains [1]. Heawood also suggested appropriate

numbers of colors for maps on a different three-dimensional surface, namely the torus. By an

extension of Euler’s formula for a torus with n number of holes, we have

F – E + V = 2 – 2n,

from which Heawood derived the formula

C(n) = [ ½ (7 + √1 + 48n)]

(where C(n) represents the number colors needed to color a

surface with n holes). Thus, for the torus above, we have n = 2 and

C(2) = [ ½ (7 + √1 + 48(2))]

= [ ½ (7 + √97)]

= [ 8.42 ]

[ 8 ]. (so eight colors are needed for a two-holed torus)

This concept was set forth in “Map-Coloring Theorems” along with Heawood’s counter

to Kempe’s proof [5]. However, even Heawood’s clever logic was not infallible; he overlooked a

vital component in presenting his own proof. While he did correctly deduce the above formula

for the number of colors needed to color an n-holed torus, he failed to show that there exists an

n-holed torus requiring C(n) colors as determined by the formula. After this rather crucial

discrepancy was exposed, his “theorem” became known instead as the Heawood Conjecture. It


was not until 1968 – seventy-seven years after “Map-Colouring Theorems” was published – that

Gerhard Ringel and Ted Youngs finally proved the hypothesis [7].

Before departing from Heawood’s work, it is interesting to note that a torus with zero

holes (or, a sphere…) requires the following number of colors from his formula:

C(0) = [ ½ (7 + √1 + 48(0))]

= [ ½ (7 + 1)]

= [ ½ (8)]

= [ 4 ].

Although a pleasing result, this unfortunately does not prove the four color theorem. It turns out

that Heawood’s formula only applies to n strictly greater than zero, as Ringel and Youngs found

out [10].

So we leave Kempe and Heawood, having seen them both produce some remarkable

results on map coloring (as well as committing a few mathematical faux pas along the way). We

now turn our attention to how their work compared to that of other map-coloring

mathematicians.

Connect the Colorists (er… dots)

Another of Cayley’s students, Peter Guthrie Tait, attempted to prove Guthrie’s conjecture

around the same time that Kempe’s proof was refuted by Heawood. Tait is probably best known

for his boasting that he had proved the theory, and not for the “proof” itself; one of his

foundational “lemma[s] easily proven” on which he based his arguments turns out to be as

challenging to prove as the four color theorem itself. However, it was Tait who introduced the

idea of coloring the actual boundary lines of regions (instead of the regions themselves) as a way

to solve the problem, and this method is still presently being considered [7].

12


Julius Petersen critiqued both Tait and Kempe’s work quite vocally, publishing his

criticism in a popular new French mathematics periodical L’Intermédiaire des mathematiciens.

He accused Kempe of “…only skim[ming] over the problem,” declaring that “…he committed

his error just where the difficulties began.” Petersen went on to say that he was not entirely sure,

but that “…if it came to a wager I would maintain that the theorem of the four colours is not

correct.” [7]

Such comments might certainly have been instigated by the fact that

Petersen was responsible for yet another graphical representation defying

Kempe’s chain argument for the hexagonal shaped v (n = 6) case. Known as a

Petersen Graph, this arrangement of eleven regions also resists Tait’s border

coloring method and serves to counter his original proof using colored borders [11]. Perhaps it is

understandable that Petersen, having come to this result and being aware of others like it, would

be skeptical of the four color theory.

Troublesome Traits

If we may be allowed a short digression, consider once more where Kempe’s proof

failed, specifically the examples that exposed its flaw. These surround his case when v is a

hexagon, bordered by five other regions; for in this instance the Kempe chains become tangled

and prevent us from four-coloring our map using his coloring method. Since this arrangement

seems so difficult to deal with, a good question to ask might be whether it is possible to

somehow represent the same situation of regions in another way that we can deal with.

German born mathematician Paul Wernicke was the first to address this question. He

followed suit behind Kempe and Heawood; his initial work on the problem was unsuccessful,

although he later produced some important results on unavoidable sets that would ultimately

contribute to solving problem. Most notably, Wernicke added two more configurations to

Petersen Graph


Kempe’s original unavoidable set, these being the cases

where two connected pentagons are surrounded by six

other regions, and where a connected pentagon and

hexagon are surrounded by seven other regions [7].

To understand why Wernicke deemed these particular

configurations unavoidable, we must first introduce the method of

discharging. A concept formulated by Heinrich Heesch, the

discharging technique allows us to illustrate the unavoidability of

many different sets. [5] To illustrate, we begin with Kempe’s first

three unavoidable configuarations (the digon, triangle, and square)

and augment them with Wernicke’s two arrangements above. If we

wish to show that these form an unavoidable set, then we can begin

with a contradiction proof by assuming that there exists a map that

does not contain regions arranged in the aforementioned ways. The

only remaining shape for a region v would then be the pentagon, as

this does not appear in our list of non-occurring configurations (at

left). So, we can have a pentagon, but it may not be bounded by

another pentagon, or a hexagon, or a digon, triangle, or square. So

what are we left with? The regions surrounding v must have more

than six edges, for such a shape is not one of our non-occurring

configurations. Thus, we have regions with seven or more edges connected to v.

Now we may assign each country on our map with E edges a “charge” of 6 – E. So, five

sided regions (pentagons) would have a charge of 6 – 5 = + 1, regions with seven edges

(heptagons) would have a charge of - 1, regions with eight edges (octagons) would have a charge

14


of - 2, and so on. (Notice that pentagons are the only regions with positive charge). If Rn

represents the number of regions with n sides, then our map has R5 pentagons, R7 heptagons, R8

octagons, etc. and we can find out the total charge of the map by multiplying the charge times Rn;

1* R5 + (-1)* R7 + (-2)* R8 + (-3)* R9 + … = R5 – R7 – 2R8 – 3R9 – 4R10 … (1)

Now we use Euler’s counting formula for cubic maps (that is, maps with exactly three

boundary lines converging at every meeting point). Using the same Rn notation above, we have:

Total # of Regions = 4R2 + 3R3 + 2R4 + R5 - R7 – 2R8 – 3R9 – … = 12. (2)

Because our map has no regions with two, three, or four edges, our formula reduces to

R5 – R7 – 2R8 – 3R9 – 4 R10 – … = 12. (3)

This equation (3) is the same as that of (1); therefore we can deduce that our map has a total

charge of +12. [5]

Now comes the “discharging.” We are able to move the charges around our map,

provided that we conserve the original amount of charge. If we consider our pentagons each with

a +1 charge, we see that we can discharge an equal amount of that charge (1/5) to each

pentagon’s five surrounding neighbors. Since all the pentagons’ neighbors must have seven or

more sides, their charges are negative integers; so adding 1/5 of a charge would keep them

negative unless they had enough surrounding pentagons contributing 1/5 positive charge to make

them positive. For a heptagon, whose charge is -1, there would need to be at least 6 surrounding

pentagons contributing 1/5 of a charge to make the charge of the heptagon positive. For an


octagon, whose charge is -2, we would need at least 11 surrounding pentagons. Not only is this

impossible (since an octagon only has eight edges), but in both cases we would have to have

neighboring pentagons – one of the configurations that cannot exist in this map.

So now we have established that, after the discharging occurs, all pentagons have zero

charge and every other region has a negative charge. However, all these negatives could never

add up to our original charge of 12! This contradiction proves that our map contains at least one

of the configurations we claimed could not be present in it. Therefore these configurations are

unavoidable, and form an unavoidable set. [7]

Keep On Colorin’

As time went on more and more configurations were discovered, and the numbers of

unavoidable sets continued to grow. Thankfully, however, it was not the sole concern of every

map coloring mathematician to disperse charges of n-gons until blue in the face. Enter George

David Birkhoff, the honey-mooning map colorist we mentioned earlier. Around 1913 Birkhoff

published a paper On the Reducibility of Maps, which generalized some of Kempe’s ideas and

extended his concept of “patching out” a map. Where Kempe reduced certain regions to points;

Birkhoff wished to reduce certain groups of regions to smaller groups of regions (hence,

reducibility) that could be easily four colored. He is recognized for his

thorough consideration of ring and diamond shaped maps, and even has

one such configuration named after him. [7]

Birkhoff’s work motivated even further exploration and construction of unavoidable sets.

Around the middle of the twentieth century, the four color theorem had gained significant

popularity outside of mathematical circles; however those working on the problem inside the

mathematical realm seemed to be themselves going in circles. Not much progress had been

Birkhoff Diamond

16


made, other than that illustrating how much more progress would be needed, as the numbers of

unavoidable sets and their configurations skyrocketed to daunting proportions.

However, in 1967, a professor at the University of Illinois named Wolfgang Haken

contacted Heesch with intentions of collaborating on the four color theorem. Haken had attended

a lecture on the progress of the problem that Heesch had given in Germany several years before,

and was interested in applying his knowledge of topology (and his stubbornness) to see where

their partnership would lead. Combining Heesch’s concept of unavoidable sets and Birkhoff’s

method of reducibility (especially of ring-shaped orientations of regions), Haken suggested a

computer might be helpful in verifying the thousands of cases for reducibility and unavoidability.

Several years passed; as they began to realize the magnitude of work to be accomplished, and

became frustrated with the shortcomings of the existing computer technology, Haken grew very

overwhelmed. While Heesch was away in Germany, Haken gave a lecture at the University of

Illinois in which he is said to have remarked that he was “finished [with the four color theorem]”

until better technology surfaced. [7]

Of all present at the lecture, Haken’s remark probably solicited the greatest reaction from

computer expert Kenneth Appel. Proficient in programming methods and navigating computer

software, Appel was quick to offer his assistance to Haken. A partnership formed immediately,

with the two men focusing mainly on fine tuning a discharging method with which to formulate

an unavoidable set of reducible configurations. Their ultimate result involved 487 discharging

rules (verified by hand-checking almost ten thousand arrangements of regions), as well as

computer testing some 2000 configurations for reducibility. [9]

The development of their solution came abruptly, both to Appel and Haken and the rest

of the world. By June 1976, the complete unavoidable set had been constructed; all that was left

was for Appel to verify it for reducibility. In checking their work, Appel and Haken solicited the


help of their families to aid in at least the proof-reading process; they knew that there were

enough stable configurations that any mistakes in reducibility would be accounted for by other

configurations. After completing the program, Appel ran it for over 1200 hours to finally obtain

a result that had escaped even brilliant minds for over a hundred years…

The four color theorem was true!

EPILOGUE

Not suprisingly this conclusion and the methods used to reach it stirred up quite a

controversy among mathematicians. Back in Germany, Heesch was a little upset that Haken had

reached a solution without him; this is understandable in light of the similarities between

Heesch’s original work and what appeared in the solution (for which he did not receive credit).

Yet this was not the only problem – others were concerned about the philosophical implications

of computer-aided proofs. Could they be trusted? Were concepts really proved if they could not

be verified by hand?

A proof is considered to consist of a finite set of axioms, from which one can deduce a

result in a finite number of steps – finite in this context often implying the ability to do it by

hand. Anyone can do a proof for himself if he does not believe it; the results are obtainable by

anyone who understands the finite axioms and steps. Certainly, Appel and Haken’s computer

completed the proof in a finite number of steps, yet it cannot be hand verified. Should we accept

proofs that cannot be performed – or confirmed – without technology? Are these really proofs?

These are the sorts of questions that haunted the success of Appel and Haken’s proof after it was

introduced. It was not until recently (1996) that another proof was formally presented, which

improved upon Appel and Haken’s methods so that only 633 reducible configurations would

need to be verified. [9] Informal proofs in the form of projects of dissertations continue to

circulate the Internet and other media discourse, yet no particular one of these appears to have

18


acquired much support. So while the original proof seems finally to have gained acceptance by

many mathematicians, the search for a simple proof - reminiscent of Kempe’s - continues.

References


[1] Calude, Andrea S. “The Journey of the Four Colour Theorem Through Time.”

University of Aukland. (New Zealand, 2001).

[2] Errera, A. “Exposé historique du problème des quatre coleurs.” Periodico di

Maternatische. (1927). Vol. 7

[3] Katz, Victor. A History of Mathematics: An Introduction. (New York: Addison

Wesley Longsman, 1998). 2nd ed.

[4] O’Connor, J.J., and E.F. Robertson. “Alfred Bray Kempe.” University of St.

Andrews, Scotland. (1996). Retrieved 10 March 2005 from http://www-

history.mcs.st-andrews.ac.uk/Mathematicians/Kempe.html.

[5] O’Connor, J.J., and E.F. Robertson. “The Four Colour Theorem.” University of St.


history.mcs.st-andrews.ac.uk/HistTopics/The_four_colour_theorem.html.

[6] O’Connor, J.J., and E.F. Robertson. “Percy John Heawood.” University of St.


history.mcs.st-andrews.ac.uk/Mathematicians/Heawood.html.

[7] Wilson, Robin. Four Colors Suffice: How the Map Problem Was Solved. (New

Jersey: Princeton University Press). 2002.

20

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Heawood.html

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Heawood.html

http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_four_colour_theorem.html

http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_four_colour_theorem.html

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Kempe.html

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Kempe.html


[8] “The Four Color Theorem.” MathPages.com. (no date). Retrieved 10 March 2005

from http://www.mathpages.com/home/kmath266/kmath266.htm

[9] Robertson, Neil, and Daniel P. Sanders, Paul Seymour, and Robin Thomas. “A Brief

Summary of a new proof of the Four Colour Theorem.” Ed. by Thomas Fowler,

Christopher Heckman, and Barrett Walls. Georgia Technical Institute. (1995).

[10] Weisstein, Eric W. “Heawood Conjecture.” MathWorld Wolfram Web Resource.

Retrieved 4 May 2005 from http://mathworld.wolfram.com/HeawoodCon-

jecture.html.

[11] Weisstein, Eric W. “Petersen Graph.” MathWorld Wolfram Web Resource.

Retrieved 4 May 2005 from http://mathworld.wolfram.com/PetersenGraph.html.

Recommended Reading

http://mathworld.wolfram.com/PetersenGraph.html

http://mathworld.wolfram.com/HeawoodCon-jecture.html

http://mathworld.wolfram.com/HeawoodCon-jecture.html

http://www.mathpages.com/home/kmath266/kmath266.htm


Appel, K. and W. Haken. “Every planar map is four colorable.” Contemporary Math. Vol

98 (1989).

N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas. “A New Proof of the Four

Colour Theorem.” American Mathematical Society. Vol 2 (1996).

Wilson, Robin. “An Update on the Four Color Theorem.” Notices of the AMS. Vol. 4 No.

7. (electronically accessible at http://www.ams.org/notices/199807/thomas.pdf)

22

http://www.ams.org/notices/199807/thomas.pdf

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