Annals of Mathematics
A Determinant Formula for the Number of Ways of Coloring a Map
Author(s): George D. Birkhoff
Source: Annals of Mathematics, Second Series, Vol. 14, No. 1/4 (1912 - 1913), pp. 42-46
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1967597 .
Accessed: 17/11/2014 06:32
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 of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of
Mathematics.
http://www.jstor.org
This content downloaded from 193.49.96.7 on Mon, 17 Nov 2014 06:32:01 AM
All use subject to JSTOR Terms and Conditions
A DETERMINANT
FORMULA FOR THE NUMBER
A MAP.
COLORING
OF WAYS
OF
D. BULXorFF.
BY GEORGE,
regionsmakingup a simply
Supposethata finiteset oftwo-dimensional
connectedclosedsurfaceare given,so thattheseforma mapM.
ormultiply
Each of theseregionsmaybe takento be limitedby closedcurves,formed
lineswhichthe regionhas in
by a finitenumberof continuousboundary
commonwithotherregions. The ends of theselines,at whichthreeor
of the map. A coloringof the map
moreregionsmeet,are called vertices
fromthat of any
to each regiona colordifferent
consistsin attributing
regionhavingin commonwith it a boundaryline, but not necessarily
fromthatofa regionmeetingit at a vertex.
different
of waysofcoloring
factwill firstbe proved: Thenumber
The following
P(X)
thegivenmap M in X colors(X = 1, 2, * *) is givenbya polynomial
of degreen, wheren is thenumberof regionsof themap M. In fact let
7n,(i = 1, 2, ** , n) be the numberof ways of coloringthe map by using
of thecolorsare disregarded.With
exactlyi colorswhenmerepermutations
it is clearthat
thisdefinition
Mi * X
(X
*
1) ..*X
i + 1)
the numberof ways of coloringthe givenmap in exactlyi of
represents
as distinctwhentheyare obtainedby a
twocolorings
the X colorscounting
onefromtheother;for,ofthei colorsused,thefirstmaybe chosen
permutation
in X ways,the secondin X - 1 ways,and so on. If X is less than i the
above termreducesto zero.
But the totalnumberof ways of coloringthe givenmap in X colorsis
the sum of the number of ways of coloringit with 1, 2, ***, n of these
the total
colors,sinceno morethan n colorscan be used. Accordingly
by
numberofwaysis represented
P(X) = MIX + M2X(X - 1) + **
X+ M"X( - 1) * *(-n
+ 1
forall valuesofX. It is clearthatin generalml = 0 inasmuchas forn > 1
no map can be coloredin a singlecolor,and that m = 1 since thereis
of the colorsbe disonlyone wayof coloringM in n colorsifpermutation
regarded.
42
This content downloaded from 193.49.96.7 on Mon, 17 Nov 2014 06:32:01 AM
All use subject to JSTOR Terms and Conditions
FORMULA FOR COLORING A MAP.
A DETERMINANT
43
In orderto proceed to the effectivedeterminationof P(X) we consider
the total number u - 1 of ways of formingfromthe map M(1) = M submaps MO), -_, M(k) of n - 1 regions,M(k+l), *.., M(z) of n - 2 regions,and
so on to M(^) of one region,by successivecoalescence of regionsadjacent
along a boundary line. Such a coalescence may be indicated by the
removal of all the common boundary lines of the two regions which
coalesce. The maps MO),
*..,
M(k)
are obtainedby one such step,the
maps M(k+l), ... , MMl)by two such steps, and so on.
At this point we introducethe symbol (i, k) to denote the number of
ways of breakingdown the map M in n regionsto a submap ofi regionsby k
simpleor multiplecoalescences,i. e., by pickingout maps M, M(GL),... , M(dk),
each but the firstbeing a submap of the preceding one, and the last one
having i regions. It is apparent that we have (i, k) = 0 for k > n -a
and that (i, n - i) representsthe numberof ways of makingn - i successive
simple coalescences. By definitionwe take (n, 0) =1 and (i, 0) = 0 fori< n.
Let now one of the X colors be placed at random on each of the regions
of any map M(i) of the ,umaps above defined. Each one of these arrangements will color one and one only of the maps M(i) and its submaps M(il),
M02)
*Y.. , namelythat one obtained by a coalescenceof all adjacent regions
whichreceivethe same colors. In consequenceif we let a-, 02, **-, a,,
denote the number of ways of coloring M(1), M(2) *.., M(^) respectively
in X colors,we will have
= 1 2, - ,)
xni
s+ 0-;+ *$(i
in whichthe symbolni denotesthe numberof regionsin M(i, and Xn,is then
the total numberof ways of giving one of the X colors to each region; on
the right appear the numbersoa, ail, oi,, .
correspondingto MWi)and its
submapsM('1),
M)
***.
Let eiifori 4 j be 1 or 0 accordingas M(i) does
or does not contain MW as a submap, and let eii be 1; we may write the
above equations in the form
VI4=
vajfj
(i =1, 2Y,**,,Y
This set of jh equations is linear in theyAquantities a,, **, , If we
observethat because of the arrangementof the submaps of M accordingto
a decreasingnumberof regionswe have i1 > i, i2>ii, ** in the above equations, it becomes clear that eii = 0 for i > j. Thus the determinantof
this systemof equations is 1 and thereforesolving foro1 = P(X) we obtain
P(X)
=
Xnly 612,
.. ..
BAA,
. .
E2n
...Y
CIn
e22X
Xn4Y& EMI2,
This content downloaded from 193.49.96.7 on Mon, 17 Nov 2014 06:32:01 AM
All use subject to JSTOR Terms and Conditions
44
GEORGE D. BIRKHOFF.
a determinantformulafor the number of ways of coloringthe map in X
colors.
This determinantis of the order u, approximatelyof the magnituden!.
Furthermoreit might be proved that the quantities esjdeterminethe constitutionof the map, so that if the determinantwere writtenin full the
structureof the completemap mightbe deduced fromit. These two facts
show the complicatedcharacterof the determinant.
The evaluation of this determinantmay be carriedout in termsof the
symbols(i, k) previouslyintroduced. To this end let us considera typical
term
VX>&2
C
...
.3
wherethe - or + signis takenaccordingas j, a, (3,***,
K
givesan odd or
i. e., accordingas the substitution
even permutationof 1, 2, **,
t1 2
i,
a
...
,uA
**-.
K
is the product of an odd or an even numberof transpositions.
Any such term eitherreduces to zero or to A">. We shall consider
how termsnot zero may arise.
If the above substitutionis not the identicalsubstitutionit may always
be decomposed into a product.of cyclicsubstitutions(P1, P2, * * *, Pk) composed of k elementsand changing P1 to P2, P2 to P3, *.**, Pk to Pl* Such a
cyclic substitutionmust contain the element 1; else there arises in the
terma product of factors
baby Ebcy
*
Ela
which is zero necessarily since we. cannot have simultaneouslya < b,
degeneratesinto a single
b < c, *... 1 < a. It followsthat the substitution
at most,containingtheelement1.
cyclicsubstitution
The correspondingterm is thus of.the form
-4-
XSj
aEab
*
'Ell (mm epp ...
wherethe + or - sign is to be taken accordingas the cyclic substitution
(1, j, a, *.A, 1) contains an odd or an even number of elements. Conversely to every product of this sort which is not zero we have a single.
termnot zero of the determinant.
Suppose now that we attempt to obtain the sum of all the terms of
this kind fora given nj = i and a given numberof elements k + 1 of this
cyclic substitution. If none of the factors Ejar Eab, ... are to be zero the
This content downloaded from 193.49.96.7 on Mon, 17 Nov 2014 06:32:01 AM
All use subject to JSTOR Terms and Conditions
A DETERMINANT
45
FORMULA FOR COLORING A MAP.
map MWj)containsM(a) as a submap, the map M(a) containsM(b) as a submap, and so on. Thus we obtain a sequence of k + 1 maps M, Mi,
M~a',***, M(M),each afterthe firsta submap of the precedingone. Consequently thereis one and only one such termcorrespondingto each way of
breaking down M in k steps to some submap of i regions,and each such
termhas the same sign as (- 1) k.
The stated terms thereforeare (_ 1)k(i, k)Xi in value and the final
formulafor the number of ways of coloringthe given map in X colors is
P(X)
n n-f
E Xi_E (
i=1
k=O
1)k(i, k).
The term in Xn,correspondingto the identical substitution,has the
unityaccordingto our previousconventionby which the
propercoefficient
symbol (n, 0) has the value 1.
As a firstexample of the formulawe take the very simplecase of a map
of threeregionswhichare adjacent each to each. We will have
(2, 1) = 3,
(1, 1) = 1, (1, 2) = 3,
and
P(X) = (3, 0)X3- (2, 1)X2+ [-(1,
1) + (1, 2)]X =
-(-
1)(X
-
2).
The validity of this formulamay be verifiedat once by noticingthat we
can color any one of these threeregionsin X colors,a second regionin the
X - 1 remainingcolors,and the thirdregionin the X - 2 colors left after
the firsttwo regionsare colored.
As a second example we take the case of a map in fiveregions formed
by a ring of three regions bounding an interiorand exteriorregion. In
this case the symbols (i, k) which enterhave the values
(3, 1) = 22, (3, 2) = 51;
(4, 1) = 9,
(2, 1) = 14, (2, 2) = 125, (2, 3) = 150;
(1, 1) = 1, (1, 2) = 45, (1, 3) = 176, (1, 4) = 150
so that
P(X)
=
x-
9X4+ 29X3- 39X2+ 18X = X(X - 1) (X - 2) (X -
3)2
In this case also the validityof the formulamay be at once verified,forthe
threeregionsof the ringmust be in threedistinctcolors,while the interior
fromthese three
and exteriorregionsmay be in any fourthcolor different
colors.
This content downloaded from 193.49.96.7 on Mon, 17 Nov 2014 06:32:01 AM
All use subject to JSTOR Terms and Conditions
46
GEORGE D., BIRKHOFF.
Even in this second case the value of the symbols (i, k) is not immediately obtained; and if we have a somewhat more complicated map, for
example the map formedby twelve five-sidedregionson the sphere,a considerablecomputationwould be necessaryto determineP(X) directlyfrom
the formula,or fromthe map itself.
PRINCETON UmvIRsITY,
May 4, 1912.
This content downloaded from 193.49.96.7 on Mon, 17 Nov 2014 06:32:01 AM
All use subject to JSTOR Terms and Conditions