Topical Paper:
The Journey of the Four Color Theorem
First Draft
by Leah Grant
for partial completion of
MATH 4010
Dr. Cherowitzo
21 April 2005
13
Chapter
Mathematicians
and 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.
3 ▪ JOURNEY THROUGH GENIUS
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.
1
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
(2)
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
2
“vertex,” and the boundaries between them as “edges.” This way
we can see the orientation of the regions and manipulate their colorings more easily.
MATHEMATICIANS AND MAP COLORING
▪ 4
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.
Francis Guthrie
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]
Frederick Guthrie
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
5 ▪ JOURNEY THROUGH GENIUS
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
Arthur Cayley
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,
MATHEMATICIANS AND MAP COLORING
▪ 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 fourcolorable. 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 R 1, 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].
7 ▪ JOURNEY THROUGH GENIUS
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:
MATHEMATICIANS AND MAP COLORING
▪ 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 fourcolor.
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
9 ▪ JOURNEY THROUGH GENIUS
and green nodes, as it would have to cross the red-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
Percy Heawood
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
MATHEMATICIANS AND MAP COLORING
▪ 10
technique. There are in fact more simplistic illustrations of such cases, involving fewer regions
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
11 ▪ JOURNEY THROUGH GENIUS
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
MATHEMATICIANS AND MAP COLORING
▪ 12
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].
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 Graph
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,
13 ▪ JOURNEY THROUGH GENIUS
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 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.
MATHEMATICIANS AND MAP COLORING
▪ 14
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
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
15 ▪ JOURNEY THROUGH GENIUS
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 Diamond
MATHEMATICIANS AND MAP COLORING
▪ 16
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
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
17 ▪ JOURNEY THROUGH GENIUS
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?
MATHEMATICIANS AND MAP COLORING
▪ 18
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
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.
19 ▪ JOURNEY THROUGH GENIUS
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://wwwhistory.mcs.st-andrews.ac.uk/Mathematicians/Kempe.html.
[5] O’Connor, J.J., and E.F. Robertson. “The Four Colour Theorem.” University of St.
Andrews, Scotland. (1996). Retrieved 10 March 2005 from http://wwwhistory.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.
Andrews, Scotland. (1996). Retrieved 10 March 2005 from http://wwwhistory.mcs.st-andrews.ac.uk/Mathematicians/Heawood.html.
MATHEMATICIANS AND MAP COLORING
▪ 20
[7] Wilson, Robin. Four Colors Suffice: How the Map Problem Was Solved. (New
Jersey: Princeton University Press). 2002.
[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/HeawoodConjecture.html.
[11] Weisstein, Eric W. “Petersen Graph.” MathWorld Wolfram Web Resource.
Retrieved 4 May 2005 from http://mathworld.wolfram.com/PetersenGraph.html.
21 ▪ JOURNEY THROUGH GENIUS
Recommended Reading
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)