ON THE FOUR-COLOUR CONJECTURE. By W. T. TUTTE. [Received 27 November 1945 Read 13 December 1945]

Transcription

1 ON THE FOUR COLOUR CONJECTURE 37 ON THE FOUR-COLOUR CONJECTURE By W. T. TUTTE [Reeived 7 November 945 Read 3 Deember 945]. Introdution The maps disussed in this paper are dissetions o suraes into simple polygons, alled regions. In eah map it is supposed that the regions are inite in number and that eah vertex o a region is ommon to just three regions. Sides and verties o the regions will be alled edges and verties o the map respetively. A olouring o a map M is deined as a set o our mutually exlusive sublasses, alled olour-lasses, o the regions o M suh that eah region belongs to some olour-lass and no two regions o the same olour-lass have any edge in ommon. I Z is a olouring whose olour-lasses are C } C, C 3 and C 4, we write The union A v B o two olour-lasses A and B o a olouring Z we all a olour-dyad. The regions o a olour-dyad may or may not orm a onneted set. In any ase we all the disjoint internally onneted omponents Kempe hains and denote their number by o {A \J B). I U is a Kempe hain o A u B and V is the remainder o A u B, then it is lear that the our sets {(AnU)v{BnV)) t ((BnU)u(AnV)), C, D, where Z = (A,B y C,D) and AnU, or example, denotes the intersetion o A and U, onstitute a olouring Z x o the map onerned and that Z x diers rom Z i and only i J o (AvB)>l. We say that Z x is derived rom Z by an exhange operation on the Kempe hain U. The set o all olourings o the map that an be derived rom Z by a inite sequene o exhange operations we all the olour-system ontaining Z and denote it by II(Z). Clearly, i Y is any olouring in U(Z), then The problem with whih this paper is onerned is as ollows. PROBLEM. Let Mbea, map on the sphere, and let M ontain a pentagon P. Let any olouring Z o that part M P o M whih is exterior to P be said

2 38 W. T. TUTTE [3 Deo. to be o type I i it has a olour-lass A whih ontains no region o M P adjaent to P, and o type II otherwise. The problem is to ind M P and Z suh that all the olourings o II (Z) are o type II. A demonstration that this problem is insoluble would omplete the veriiation o the our-olour onjeture by enabling us to dedue rom the existene o a olouring o M P the existene o a olouring o type I o M P and thene (by assigning Pto A) the existene o a olouring o M. The ontribution o this paper to the problem is the dedution o some new limitations on the struture o II (Z) whih must be satisied in any solution that may exist. Use is made o an elementary general theorem on the olourings o spherial maps, to whih I have not seen any reerene in the literature. This is the "Parity Theorem" proved below. Reerene may be made to a paper by KittelJ on the above problem. In omparing this paper with his it should be borne in mind that he ounts as distint olourings whih dier only by a redistribution o "olours" among the olour-lasses a distintion whih has no meaning with the deinitions used here.. The parity theorem Let Z = (A, JB, C,D) be a olouring o the spherial map M o a regions. Let o (X) and a, (X u Y) denote the number o regions in the olour-lass X and the olour-dyad (X u Y) respetively, and let ti(x u Y) be the number o edges in whih regions o X meet regions o Y. Then, i x (X \J Y) is the onnetivity o the set o regions XuY, we have by elementary topology CQ( A O B)-C {A KJB) = a (A v B)~p x {A u B) () and C (AVB) = C (CKJD)-. () From these equations and the orresponding ones or the olour-dyads AKJC and A u D it ollows that o (A u 5 ) + o {A vc) + o (A u D) - Q (C u D) - o (B u D) - o (B u C) = 3a (A) + a (B) + a (C) + <x (D)-^(A^B)-i {AKjC)-i (^^D)-^ 3, (3) RA where (R) denotes the number o sides o the polygon R. Heawood, Quart. J. o Math. 4 (89), % Kittel, Bull. Amerian Math, So. 4 (935), See, or exariaple, Newman, Topology o Plane Sets (C.U.P. 939), The regions must be disseted into triangles beore his results are applied, but this introdues no real diiulty.

3 945] ON THE FOUR COLOUR CONJECTURE 39 Now the right-hand side o (3) depends only on the olour-lass A. Hene we have THEOREM I. The quantity on the let o equation (3) has the same value or all olourings o M or whih A is a olour-lass. We say that two olourings Z x and Z % o a map are onneted i there exists a inite sequene o olourings o the map beginning with Z x and ending with Z and suh that eah pair o onseutive members have a olour-lass in ommon. Clearly any two olourings whih belong to the same olour-system are onneted. (But the onverse does not ollow.) For any olouring Z, let J(Z) denote the sum o the quantities o (X u Y) over the six olour-dyads. We all the parity o J{Z) the.parity o Z. THEOREM II (the parity theorem). / / Z x and Z are onneted olourings o a spherial map M, then For by (3), or any olouring o M J{Z)= ReA, (mod ). (4), (mod), where A is any olour-lass o Z. Hene (4) is true whenever Z x and Z % have a olour-lass in ommon, and thereore it is true whenever Z x and Z % are onneted. It may be worth mentioning that the olourings o a partiular spherial map need not be restrited to one parity. For example the two olourings shown in Fig. have opposite parities. A B / \ A A\ D \ \ D \ VA X B \ / C / \ \ D A X A \ X \ \ B VA / D A Fig. B

4 4 W. T. TTJTTE [3 De. 3. The pentagon problem Let us now return to the problem o the partially oloured map stated in the introdution. Let F denote the set o regions o M whih meet P. We denote these regions by F v F, F s, F 4, F 6 in the yli order o inidene with the edges o P. We may suppose that they are taken in lokwise order as seen rom the entre o the sphere. We have not yet assumed that they are all distint. In a olouring o type II eah olour-lass ontains a region o F. Three olour-lasses thereore eah meet P in one edge only, and the ourth meets it in just two edges, E x and E say. E and E do not meet, or i they did the third edge inident with their ommon vertex would be ommon to two regions o the ourth olour-lass. There is thereore an edge E z o the pentagon adjaent to both E x and E. We all the set o regions o F whih have E or E as an edge the norm o the olouring, and the region o F having E z as an edge the apex o the olouring. The ollowing well-known result will be needed, THEOREM III. I Z is a olouring o M P suh that all the members o II(Z) are o type II, then i any Kempe hain ontains the norm o Z it ontains also the apex o Z. For let Z = (A, B, C, D) where A ontains the norm and B the apex o Z. Then i the theorem is alse or Z one o the remaining two regions o F, e C say, must be separated in M P rom the apex by a Kempe hain o A u D ontaining the norm. Hene, by an exhange operation on that Kempe hain o B u C whih ontains the apex, a type I olouring o II (Z) an be obtained, ontrary to hypothesis. It ollows rom theorem III that i all the olourings o U(Z) are o type II, then the ive regions F t are all distint. We thereore assume their distintness in what ollows. I F is the apex o Z, then the norm o Z is the pair o regions F i+li F i+^. (The addition in the suies is modulo 5.) We then denote the Kempe hain ontaining F i+ and F i+ by g(z) and the olour-dyad to whih it belongs by O(Z). We denote the Kempe hain o O(Z) whih ontains the other member F i+i o the norm by h(z). It ollows rom theorem III that g(z) and h(z) are distint. We deine a X-operation on Z as the appliation o the exhange operation to eah member o a subset A o the Kempe hains o G(Z), where A ontains g(z) but not h(z). Heawood, lo. it.

5 945] ON THE FOUE COLOUR CONJECTURE 4 (Note. A similar operation on O(Z) aeting h(z) but not g(z) would be equivalent to the A-operation aeting just those omponents o O(Z) whih the ormer operation let unhanged.) We deine a X-iruit in II (Z) as a yli sequene o olourings belonging to IT(Z) suh that eah member o the sequene an be derived rom its immediate predeessor by a single A-operation. A-operations and A-iruits an o ourse be deined in the same way when II(Z) ontains type I olourings but they are o partiular interest in the other ase, or then it has been shown that every olouring in U(Z) is a member o a A-iruit. I a A-operation is applied to a olouring Z o apex F i the norm o the new olouring will be the pair F i+, F i+4, and its apex will thereore be F i+3. The olour-lasses o the old and new apies must be ommon to both olourings, sine neither belongs to (Z). Eah A-operation thus advanes the apex three plaes in the yli sequene o the F t. Hene i the number o members o any A-iruit is n, then n =, (mod 5). (5) The objet o this paper is to improve upon this result by establishing the ollowing two theorems: THEOREM IV. =, (mod ). THEOREM V. 4. Proo o theorem IV Let XZ be the olouring obtained rom Z by appliation o the A-operation A. The intersetions o the olour-lasses with F will be as shown in the diagrams (i) and (ii) o Fig. or Z and XZ respetively. The olour-lasses B and D are ommon to both olourings. Fig. t Errera, "Du Coloriage des Cartes et de Quelques Questions d'analysis Situs", Thesis (Gauthier-Villars, 9). See also Kittel, lo. it.

6 4 W. T. TXJTTE [3 D e Sine the regions o F are distint, a new spherial map N an be ormed rom M by inorporating P in that region o F whih belongs to B in the olourings Z and XZ. I the new region is assigned to B, the olourings Z and XZ will be transormed into olourings Z N and (XZ) N respetively o N. These two olourings o N are onneted, or they have the olour-lass D in ommon. Hene by the parity theorem J(Z N ) = J((XZ) N ), (mod ). (6) Now in the hange rom Z to Z N the only two o the six quantities o (X u Y) not obviously unaltered are o (B u C) and o {B u D). But or one o these, let us say o (B u C), to be altered it is neessary or the two regions in whih the orresponding olour-dyad o Z meets F to be in dierent Kempe hains o that olour-dyad. This olour-dyad (B u C) o Z would then ontain the apex and would not separate the members o the norm in M P. The members o the norm would thereore belong to the same omponent o {A u D) in Z, whih ontradits theorem III, sine the apex o Z is in B and thereore not in (A u D). Hene J(Z N ) = J(Z). (7) Again in the hange rom XZ to (XZ) N the only quantities Q (X v Y) not obviously unaeted are o (B u D) and o (B u A'). But the irst o these is unhanged, by the last paragraph, sine B \J D is the same in Z as in XZ and in Z N as in (XZ) N. However, by theorem III the two members o the norm in XZ are in dierent Kempe hains o (B v A') (whih does not ontain the apex). o (B u A') thereore dereases by and so From (6), (7) and (8) we have l. (8) J{XZ) = J{Z) + l, (mod ). (9) It ollows that the number o members o any A-iruit must be even. Hene by (5), ^ ( m o d l ) ( ) 5. Proo o theorem V Assume that a A-iruit o members exists. Let its members be in order Z ) Z x, Z,..., Z 9. Let A i denote that olour-lass o Z i whih ontains the apex. By the paragraph immediately preeding equation (5), A i-x, A t and A i+ are all olour-lasses o Z t. Moreover they are distint, or the apies o Z t _ v Z i and Z i+ are distint regions F it F i+z and F i+. (Addition in the suies o the A i and the Z i is mod.)

7 945] ON THE FOUR COLOUR CONJECTURE 43 The ten olourings Z i are thereore ompletely determined when the ten olour-lasses A i are given, or when three olour-lasses o a olouring are given the ourth is uniquely determined. I R is any region o M F we write v { = v t (R) = i R is in A it = i R is n o t i n g. () Then the set o -vetors V(B)^{v o (B),v (B),... t v 9 {B)) given or all R ompletely determines the A t and thereore the Z v As an example we note that F t is in A i i and only i it is the apex o Z i and that this happens just one in any ive onseutive members o the A-iruit. So we may write V(F X ) = ( ), () a vetor whih we denote hereater by e. We then have, by the properties o A-operations, and in general, V(*i+r) = Q*e* ( 3 ) where Q is a yli permutation deined by Q(v o,v v...,v 9 ) = (v 9i v,v v...,t> 8 ). (4) We denote the group o yli permutations Q* by Q, and say that two -vetors are equivalent when they an be transormed into one another by operations o &. A -vetor whose omponents are restrited to the values and we all a V-vetor. We say that a F-vetor F is admissible i it satisies the ollowing onditions: (i) no three onseutive signs o F (regarded as a yli sequene) inlude more than one ; and (ii) there exist three other F-vetors satisying (i) suh that their sum with F is J = ( ). The other three F-vetors o (ii) are learly admissible. We all a set oi our admissible F-vetors whose Sum is / a tetrad. We have at one LEMMA I. / / V and V are equivalent V-vetors and V x is admissible, then V is admissible. For the properties (i) and (ii) are invariant under the transormation Q. LEMMA II. / / R is any region o M P, then V(R) is an admissible V-vetor. For irst, by (), V(R) is a F-vetor.

8 44 W. T. TT7TTE [3 De. Seondly, suppose two o v^r), v^r), and v i+ {R) have the value. Then, by (), R belongs to two o the olour-lasses A^, A iy A i+ whih is impossible sine these are distint olour-lasses o the same olouring Z {. Hene V(R) satisies (i). Thirdly, any region R o o M P has some vertex not a vertex o P. For i this were alse or Fj then Fj_ x and F j+ would not be distint. At this vertex R o meets two other regions, R and R say, o M P. Consider the matrix whose three rows are the vetors V(R ), ViRJ, V(R ). No olumn o this matrix ontains more than one, sine no two o R o, R and R belong to the same olour-lass in any olouring o M P. Hene V = I- ViRJ-ViRJ-ViRJ is a F-vetor. Consider the (i l)th, ith and (i + l)th olumns (addition mod ). I two o them onsist entirely o O's, then Z i has a olour-dyad ontaining none o the regions R o, R ± and R by (). But this is impossible or it requires that two o these mutually ontiguous regions shall belong to the same olour-lass o Z t. It ollows that the F-vetor V satisies (i) and so by the previous result that V(R) is a F-vetor satisying (i), and by the deinition o F', it ollows that V(R ) satisies (ii). This proves the lemma. COROLLARY. / / three regions R o, i^ and R o M P meet at a vertex, then the our vetors F(J? ), F ^ ), V(R ), I-V^-ViRJ-ViRt) are admissible V-vetors and onstitute a tetrad. I F is a F-vetor, we denote by «r( F) the number o its 's. By onsidering in turn the ases r(f) =, o~(v) =, et., we ind that every F-vetor satisying (i) is equivalent to a member o the ollowing set. a = ( ), 6 = ( ), = ( ), d = ( ), e = ( ), /= ( ). (5) Clearly no two members o this set are equivalent. I x is one o the vetors satisying (i) we deine m(x) as the smallest integer m not suh that Q^x = x. We then have m(x) = i x is equivalent to a, = 5 i x is equivalent to e, = otherwise. (6)

9 945] ON THE FOUR COLOUR CONJECTURE 45 The next step in the proo is the determination o all the tetrads. We note that i the vetors V i (i =,,3,4) onstitute a tetrad, then X^ = (7) i by the deinition o a tetrad; and or eah riy^ (by (5)) O^F^. (8) The only sets o our integers satisying (7) and (8) are (3,3,3,) and (3,3,,). (9) The F-vetor a thereore is not admissible. Now the only vetor x o (5) or whih <x(x) = is 6 and the only one or whih r(x) = 3 is/. Hene by (9) any tetrad involving 6 is o the orm Now / ontains just one blok o three onseutive 's. Consider the matrix whose rows are the vetors o T. The three olumns whih ontain the orresponding blok in Q*/must ontain three l's altogether (deinition o a tetrad) and this is possible only i they ontain the non-zero omponent, o b (by (i)) whih implies that i =,, or 3. The same argument applies with i replaed by j or k. Hene i b is ontained in a tetrad, that tetrad is ^ and it is easily veriied that these our vetors do indeed onstitute a tetrad. I we apply an operation Q i to eah o the our F-vetors o any tetrad T we shall learly obtain a new tetrad. We denote this by Q l T and say that it is equivalent to T. In listing the tetrads it will suie to give one member o eah set o equivalent tetrads. We may thereore proeed to those tetrads whih ontain no vetor equivalent to 6. By (5) and (9) suh a tetrad involves just two vetors equivalent to /, and is thereore equivalent to a tetrad involving a pair/, Q>j. We an urther suppose,; less than 6, or the operation Q x - j transorms the above pair into the pair/, Q ~ j (by (6)). By omparing the irst o the six F-vetors/, Q, Q, Q 3, Q*, Q 5 with the other ive, we ind that the ases j = 3 and j =.4 are impossible sine or eah o these there is an s suh that v 8 () = v 8 {Qi) =, ontrary to the deinition o a tetrad, but that the other ases annot be ruled out in this way. We may suppose thereore that j =, or 5. Now I--Q= ( ) = QW+Q*d = Q 5 +Q 9, SEB.. VOL. 6. NO K

10 46 W. T. TUTTE [3 De. and these are the only two ways in whih the vetor ( ) an be expressed as the sum o two vetors equivalent to members o the set (5) and satisying <r(z) =. Again / - / - Q = ( ), whih an be expressed subjet to the above onditions only as Finally / - / - Q 5 = ( ), whih must be expressed either as Q e+q*e or as Q* + Q 9. Every tetrad thereore is equivalent to a member o the ollowing set: Zi = (6, Q, Q% Q 3 )> ) T = (, Q, Q% Q»d), T 3 = (, Q, Q% Q»), T, = (, Q, Q 7 d, Q*e), T 5 = (, Q 5, Q e, Q'e), T, = (, Q 5, Q% Q 9 ). () The signiiane o the set W o all admissible F-vetors an best be understood in terms o the dual map M* o M. The regions o M* are all triangles and so M* is a simpliial dissetion o the sphere. I R is any region o M, we denote the orresponding vertex o M* by R*. The dual map M* o M P may be deined as the set o all simplexes o M* whih do not have P* as a vertex. It is thereore a simpliial dissetion o a part o the sphere bounded by a simple losed urve F* whih onsists o the verties Ft, F$,..., Ft and the edges F$F%, F^F^,..., F;F dual to the ive distint edges o M P whih meet the pentagon P. (I there were not ive suh distint edges o M P, some two verties o the pentagon would be joined by an edge E in M P and then at least one o the two regions o M P inident with E would be inident with two o the edges o P. This would ontradit the requirement that the ^ must be distint.) It ollows that the ormal sum S (*?, i^-u) = K say, () i = l where (F, F + ) is a -simplex o F* with an orientation given by the order o the t rms F and F +V is a bounding -yle on M%.

11 945] ON THE FOUR COLOUR CONJECTURE 47 We an treat W as a simpliial 3-omplex o whih the F-vetors are the O-simplexes and the tetrads the 3-simplexes (and in whih eah i-simplex is inident with an (i + l)-simplex or i < 3). The orrespondene R* -> V(R) deines a mapping o M* into W whih maps verties on to verties, and, by the orollary to lemma II, -simplexes on to -simplexes. It ollows at one that i the orrespondene maps the -hain U o M% on to the -hain U w o W, where U and U w have ordinary integers as oeiients, then it maps the boundary o U on to the boundary o U w. Consequently bounding yles on M* are mapped on to bounding yles on W. Now the bounding -yle K on M% maps on to a -yle K w =, V(F M )) by () = ((e, Q e) + (Q\ Q*e) + (Q*e, Qe) + (Qe, Qh) + (Qh, e)) by (3) and (6): But it an be shown that this is a non-bounding yle o W. By proving this we shall show that the hypothesis o the existene o a A-iruit o members leads to a ontradition and so establish theorem V. To do this we irst deine a untion o the equations or eah -simplex (V^V^) o W by means ^ ( ) and the ollowing table. TABLE I (3) Reerene number v x v, A(F X, F ) Reerene number F x F A(F S F ) b b b 'd d Q Q J Q 3 Q 5 Q Q. Q* Q«Q*d Q e d d d d e e e Q / Q 3 Q* Q* Q e Q Q 3 Q Q Q b - It may readily be veriied (with the help o (6)) that no two o the pairs o this table, even when regarded as unordered, are equivalent under the operations o. The deinition o A(P^,T^) is thereore onsistent. K

12 48 W. T. TUTTE [5 Nov. Other assertions veriiable rom the table with the help o equations (6), () and (3) are: (i) A(T,F ) is deined or every -simplex (^,F ), that is or every ordered pair o F-vetors both ontained in the same tetrad; and (ii) i V v V and V z are distint members o the same tetrad, then It is suiient to veriy these two assertions or eah o the tetrads o () by (3). This veriiation is set out in tabular orm in Table II. The numbers in the last olumn o this table are the reerenes, in order, to the rows o Table I. Equation (4) states that the sum o the untion A^,!^) over the boundary o any -simplex o W is. It ollows rom this, and () that the sum o the untion over any bounding -yle o W is. But its sum over the -yle K w is, by (3) and the preeding expression or K w, 5A(e, Q e) = (by Table I). Thus K w is shown to be non-bounding and the proo o theorem V is omplete. TABLE II Tetrad F x F a F 3 A(F l F a ) A(F, F 3 ) A(F 8, V x ) Sum Reerenes T x T, T> T t T> 6 b b Q Q Q QV Q* / / / Q* Q Q Q Q Q Q Q*d Q*d O Q Q 5 Q 6 Q* Q* Q 7 d Q'd Q* Q 6 Q a e Q*e Q' Q' Q Q* Q* Q* Q 6 d Q 8 d Q*d Q*d Q 6 Q 7 d Q l e Q'e Q*e Q*e Q'e Q'e Q'e i j - - -, 9,, 9, 3,, 3 9, 9, 9, 6, 4 9, 3, 4,, 5,, 3 9, 9, 8 9, 7, 6 8, 4, 6 9, 4, 7, 4, 3, 8, 7 3,, 7 4,, 8, 8, 8, 7, 7 8, 6, 7 8, 6, 7, 6, 9, 9, 6 9, 5, 6 6, 5, 9

13 945] ON THE FOUR COLOUR CONJECTURE 49 The simple method used in the proo o theorem V will not suie to prove the analogous theorem or n =. For Errera has given a map M P and a olouring Z suh that a ertain sequene o A-operations transorms Z into itsel. Perhaps it is signiiant that this map ontains a region (the entral one in Kittel's diagram) whose -vetor V satisies Q 4 (F) = V, where Q and F are deined by equations analogous to (4) and () respetively. Even in this ase IT (Z) ontains olourings o type I. Trinity College Cambridge Errera, lo. it. and Kittel, lo. it.