IIn
1890, Percy J.Heawood produced a map forwhich Kempe's
process would fail. Heawood's example revealed a subtletythat
had escaped detection by the restof themathematics community
Altred
Kempe's
of
Bray
"Proof"
the
Theorem
yTIMOTHYSIPKA
Alma College
Twenty
y
have passed
and
Haken
since Wolfgang
Kenneth Appel provided the
mathematics communitywith a proof of
thewell-known theorem that any map
on a plane or surface of a sphere can be
colored with atmost four colors so that
no two adjacent countries have the same
color. Their conquest of the four-color
theorem came almost a century after the
world had accepted the first"proof of
the theorem. In 1879, Alfred B. Kempe
published what he and themathematics
community thoughtwas a proof of the
four-color theorem. Unfortunately for
laterP. J.Heawood
Kempe, eleven y
discovered a flaw.This articlewill take a
close look at Kempe's
attempt to prove
the four-color theorem. In addition, we
will discuss the conjecture's origin and
consider Heawood's
counterexample
thatexposed the flaw inKempe's work.
and found no tendency by
mapmakers tominimize the number of
Congress
Sharing the conjecture with De
Morgan was a stroke of luck for the
colors. May found very few maps that
used only four colors, and those thatdid
usually required only three colors. May
mathematics
concluded, "if cartographers are aware
of the four-color conjecture, they have
certainly kept the secretwell."
So when did the four-color conjec
ture actually arise? Some believe that
Guthrie, a British mathemati
was
the first person tomake the
cian,
conjecture. In fact,May believed the
y
conjecture "flashed across themind of
Guthrie while he was coloring a map of
y
y
We
do know that in 1852
had a conversation with his
inwhich he stated and
a
attempted
proof of the conjecture.
y
mentions thaty
"showed
me the fact that the greatest necessary
brother y
number of colours to be used in colour
ing a map so as to avoid identity of
colour in lineally contiguous districts is
went on tomention that
four." y
Originof the Conjecture
It has been conjectured that early map
makers were the firstto notice that four
map.
gave "did not seem
altogether satisfactory to himself,"
which probably explains why y
never published it. Soon after this con
support it. In the early 1960's, Kenneth
May reviewed a sample of atlases in the
shared the conjec
versation, y
ture with Augustus De Morgan, his
mathematics professor at University
suffice when coloring a
This claim, though logical and
tempting tomake, has little evidence to
colors would
large collection
of
the Library
of
the proof y
College London.
community De Morgan
immediately began to make inquiries
about the problem. In an letter to
William R. Hamilton, dated October 23,
1852, we have the firstwritten reference
to the conjecture. In that letter, De
Morgan asked whether Hamilton had
heard
the conjecture. Hamilton
promptly replied that he had not, and
that he would not likely attempt the
of
problem. We know of two other letters
fromDe Morgan inwhich he discussed
the four-color conjecture. In the first let
ter,dated December 9,1853, De Morgan
wrote to his former teacher, William
Whewell;
24,
y
and in the second, dated June
1854, he wrote
to Robert Leslie
In addition to spreading news of the
conjecture through lettersand conversa
tions, De Morgan was also responsible
forwriting the first article that referred
to the conjecture. In a book review of
Whewell's
The Philosophy ofDiscovery
in
theApril 14, 1860 issue of
published
theAthenaeum, De Morgan included a
paragraph that described the four-color
conjecture. The paragraph also con
tained the comment, "itmust have been
alway
known
to map-colorers
different colors
that four
are enough" which,
WWW.MAA.ORG 21
MATH
quite possibly initiated the tradition of
linking the four-color conjecture with
cartographers.
1860 and 1878, interest in
the four-colorproblem appeared towane,
Between
and according to Rudolf and Gerda
"the problem was not discussed
y
in themathematical literature
any
of the time." However, at a meeting of
the London Mathematical
Society on
June 13, 1878, Arthur Cay
appeared
to revive interest in theproblem when he
had proved the
inquiredwhether any
He
then
conjecture.
published a short
paper inwhich he gave a precise state
ment of the conjecture and explained
where the difficultywas inproving it.
Itwas not long afterCay
's
inquiry
thatAlfred Bray Kempe, a London bar
rister and former student of Cay
arrived at his now famous and fallacious
proof of the four-color conjecture. News
of his "proof was announced in the July
17, 1879 issue ofNature, while the full
text appeared shortly thereafter in the
recently founded American Journal of
Mathematics.
The "Proof
There are two observations that should
be made when one reads Kempe's paper,
observations thatmay explain why the
subtle error in his argument went unde
all of his
tected for eleven y
y
are
are
16)
diagrams (there
relatively
simple, and most of them are used to
provide examples of the terms he
defines. He never provides a nontrivial
diagram (map) that demonstrates his
argument. Second, thepaper is virtually
all prose which, though well written,
makes itdifficult to verify his work.
Though the phrase "mathematical
induction" was never mentioned
in
Kempe's paper, the "patching process"
he used made his argument essentially a
induction.
by mathematical
Therefore, in presenting Kempe's argu
proof
As with most induction proofs, the
base step is quite obvious: anymap con
taining four or fewer countries can easi
four colors, and then letM be a map
consisting of n + 1 countries. It can then
be shown?and
2002
R
did so?that M
Kempe
must contain at least one country that is
adjacent to five or fewer other countries.
Let X denote such a country inM; then
temporarilydisregardX. We are leftwith
is adjacent to exactly five
countries colored with four different
colors.
^^^^^
In handling these two cases, Kempe
used a technique that todaywe call "the
method ofKempe chains." He firstasked
thatwe consider all the countries (he
called them districts) in themap which
are colored
a map of n countries,which we'll denote
red and green; then he
observed that these countries formone or
byM-X. Now, color the countries ofM
Xwith atmost four colors. Let's use red,
more red-green regions. Kempe's notion
of a red-green region was simply a con
as Kempe did.
"take a piece of
said
Kempe actually
cut
out
to
and
it
the same shape"
paper
's,as the countryX9 and then "fasten this
blue, green, and y
patch to the surface and produce all the
boundaries which meet thepatch tomeet
at a point within the patch." In other
words, Kempe described a process that
phy
extended
removed the countryX and
the boundaries of the sur
rounding countries tomeet at a point
within the region once covered byX.
In themap M-X, we have colored n
countries with at most four colors, and
we've leftX uncolored. Kempe's goal
to find a way to reduce (if neces
sary the number of colors used to color
was
the countries surroundingX so that some
color would be "free" forX. He quickly
ifX
dispensed with the easy cases. y
is surrounded by three or fewer coun
tries, then clearly therewill be a color
available forX. Second, ifX is sur
rounded by four or five countries col
ored with atmost threecolors, then there
will also be a color available forX With
these cases out of theway Kempe was
leftwith two cases to consider:
Case
tinuous "chain" of countries colored red
or green. He thenmade the important
observation that one could interchange
the colors in any red-green region, and
themap would still remain properly col
ored. We will now demonstrate, using
nontrivial
the arguments
examples,
cases.
two
the
for
gave
Kempe
y case 1,we firstlabel the countries
surroundingX with the lettersA, B, C,
and D. Kempe then considered two sub
cases.
Subcase 1.1: Suppose countries A and
C belong to different red-green regions.
1we have an example of a
in
which
countries A and C belong
map
todifferentred-green regions. In this sit
In y
uation, Kempe observed that "we can
interchange the colours of thedistricts in
one of these regions, and the resultwill
be that the districts A and C will be of
the same colour, both red or both green."
By interchanging the colors in the region
2 that
containing A, we see in y
both A and C are now green,making the
color red available forX.
1: X
is adjacent to exactly four
countries colored with four different
colors.
3
NOVy
Case 2: X
ly be colored with at most four colors.
Now, assume thatanymap containing n
countries can be colored with at most
ment, we will use his vocabulary and
basic ideas, but we'll give a more con
temporaryversion of his proof.
22
HORIZONS
?B
ALy
BRAY
D Ky
'S
*TROOy
Subcase
1.2:
and C belong
Oy THyy
THy
M
Suppose countries A
to the same red-green
region.
3 we have an example of a
in
which
countries A and C belong
map
to the same red-green region. Kempe
In y
ft
observed in this case that the red-green
y
2
region will "form a ring" preventing B
and D frombelonging to the same blue
region. Therefore, by interchang
ing the colors in exactly one of these
blue-y
regions, we reduce to three
y
7
y
the number of colors surroundingX. In
4 we have interchanged the col
y
ors in theblue-y
region containing
B, making the color blue available forX.
case 2,we label the five countries
surroundingX with the lettersA, B, C,
y
y
3
so that the
blue-green region towhich B belongs is
different from that to which D and y
belong, and the blue-y
region to
which y belongs is differentfrom that to
which B and C belong." To reduce the
number of colors surroundingX, Kempe
thenmade the claim, "interchanging the
we
have
Suppose
either countries A and C belonging to
differentred-y
regions or countries
colours in theblue-green region towhich
B belongs, and in theblue-y
region
towhich D belongs, B becomes green
and y y
A, C, and D remaining
and y Kempe
Subcase
2.1:
A andD belonging to differentred-green
regions.
When
4
two regions cut offB from y
then considered two
D,
subcases.
y
same red-green region. In a case such as
this,Kempe correctly observed that "the
one of these alternatives
is
present in a map, we simply perform an
interchange of colors similar to the
8, the interchanges
unchanged." In y
of colors have been performed as Kempe
described with the outcome he expected,
making the color blue available forX.
5
process used in subcase 1.1. In y
we have an example of a map inwhich
countries A and C belong to different
regions. Then, as Kempe
claimed, "interchanging the colours in
or
either,A and C become both y
both red." Ifwe interchange the red and
red-y
colors in the region containing y
y
we obtain the coloring iny
6,mak
y
ing red available forX.
y
5
Subcase 2.2: Suppose countries A
and C belong to the same red-y
region and countries A and D belong to
the same red-green region.
In this, the fourth and final case,
Kempe's process for reducing the num
ber of colors surroundingX contained a
subtle flaw. In y
7 we have an exam
y
6
ple of a map where countries A and C
belong to the same red-y
region and
where countries A and D belong to the
Heawood's
8
Counterexample
In the example used in subcase 2.2,
Kempe's process worked exactly as he
had hoped. By simultaneously inter
changing the colors in the blue-green
region containing B and theblue-y
region containing y the number of col
ors surroundingX was reduced to three.
Unfortunately forKempe, this process
Continued on p. 26.
WWW.MAA.ORG 23
MATH
HORIZONS
Continued from p. 23.
would not work for all maps
Conclusion
The importance ofKempe's work cannot
be overlooked. His basic ideas provided
the starting point forwhat would be a
satisfy
the conditions of subcase 2.2.
In 1890, Percy J.Heawood produced
a map forwhich Kempe's process would
fail.Heawood's example revealed a sub
tlety that had escaped detection by the
restof themathematics community And
that subtletywas the possibility that the
blue-green region containing B and the
region containing y might
blue-y
"touch." When
this happens, Heawood
observed, "y
transposition prevents
the other frombeing of any avail."
9 we see themap Heawood
to expose the flaw in Kempe's
In y
used
process for reducing thenumber of colors
in subcase 2.2. Notice that theblue-green
region containing B and theblue-y
region containing y share a boundary If
century of effortculminatingwith Appel
and Haken's proof. In 1989, as a tribute
y
9
we interchange the colors inboth regions,
the two countries sharing thisboundary Y
and Z, would both receive the color blue.
as Heawood
toKempe, Appel
and Haken declared:
argument was
extremely
"Kempe's
and
clever,
although his "proof turned
out not to be complete, itcontained most
of the basic ideas that eventually led to
the correct proof one century later."
remarked, "Mr.
Kempe's proof does not hold unless some
modifications can be introduced into it to
y
meet this case of failure."
Interested readers will
Thus,
Kempe certainly tried to fix this "case
of failure," but neither he nor any of his
contemporaries could do so. The modifi
cations thatwere needed would require
many y
ofwork bymany individuals.
y
Reading
find a detailed
history of the four-color problem and a
thorough list of the relevant literature
in The y
Theorem: History
and Idea of
y
Topological
Rudolf
and
Gerda
y
Proof by