irving kittell- a group of operations on a partially colored map

8/3/2019 Irving Kittell- A Group of Operations on a Partially Colored Map

1/7

1935-1 COLORED MAPS 407

A G R O U P O F O P E R A T I O N S O N A P A R T I A L L YC O L O R E D M A PBY IRVING KITTELL

I t has been p rev ious ly shown* tha t i f the re be a min imumunco lo rab le map , and i t i s co lo red wi th the excep t ion o f onefive-sided region in the four colors A, B, C, and D, then on lyone of the colors (say B) will be repeated in the f ive regionstouch ing the reg ion no t co lo red , and tha t th i s r ing o f r eg ions canbe ass igned the colors DBABC in order . I t has l ikewise beenshown tha t when the map i s thus co lo red ,the re wi l l be two in te r sec t ing Kempe cha insrunn ing f rom A to C a n d A to D y respec t ive ly .W e s h a ll s a y t h a t a n y m a p , w h e t h e r m i n i m u muncolorable or not , which is par t ia l ly coloredin the above manner i s co lo red impasse. T h es i tua t ion may be rep resen ted as in F ig . 1 . FIG. 1T h e d i a g r a m d o e s n o t a t t e m p t t o s h o w wh e r e o r h o w m a n yt imes the c i rcu i t s c ross one ano the r . I t mere ly means tha t theydo cross . Here we have placed the uncolored f ive-s ided regionou ts ide the res t o f the map . We do th i s fo r compac tness , bu t i tmus t be no ted tha t wha t fo l lows i s no t dependen t upon th i schoice of representa t ion , nor upon the choice of le t ter ing . Alsoto be remembered i s the fac t tha t r igh t and le f t have on ly a rb i t r a ry mean ings in d i scuss ing a map which may be de fo rmed inany con t inuous manner over the su r face o f a sphere . Mos twr i te r s would d raw such a map wi th the unco lo red reg ion in s ide . We here ca l l the D reg ion the one in the d iagram above onthe r ight of A , and the C region the one on the left of A . W emake the fo l lowing def in i t ions .The reg ion A is called the vertex of the r ing. I t is the regionin th i s r ing which l i e s be tween the two reg ions hav ing the samecolor . We cal l the AC c i rcu i t the left-hand circuit a n d t h e ADc i rcu i t the right-hand circuit. We ca l l the BD chain which inc l u d e s t h e B region in the r ing between A a n d C t h e left-hand

* P. J. Heawood, M ap colour theorem, Quarterly Journal of Mathematics,vol. 24 (1890), pp. 332-338.


2/7

408 IRVING KITTELL [June,chain a n d t h e BC cha in which inc ludes the o ther B th e right-hand chain. We ca l l the CD cha in which inc ludes the C an d Dregions of the r ing the end tangent chain; t h e BC cha in touchingthe r ing in the ad jacent B an d C regions the left-hand tangentchain ; and the BD cha in touching the r ing in the ad jacent B a n dD reg ions the right-hand tangent chain. Final ly , we cal l the ABcha in which inc ludes the ver tex , the osculating chain.We now d is t inguish n ine opera t ions tha t can be per formed ont h e m a p a b o v e .a : t ranspos ing the co lors o f the le f t -hand cha in ./3: t ranspos ing the co lors o f the r igh t -hand cha in .7 : t ranspos ing the co lors o f the le f t -hand c i rcu i t . : t ranspos ing the co lors o f the r igh t -hand c i rcu i t .e: t ranspos ing the co lors o f the end tangent cha in .f : t ranspos ing the co lors o f the le f t -hand tangent cha in .77: t ranspos ing the co lors o f the r igh t -hand tangent cha in .0: t ranspos ing the co lors o f the oscu la t ing cha in .1: l eav ing the co lor ing unchanged .In al l of these operat ions no color a t a l l wil l be assigned to theouts ide reg ion .Now, un less each of these opera t ions or any combina t ion ofthese opera t ions leaves the map in a form ana logous to the f i r s tform abo ve ( th a t i s , wi th tw o in te rsec t ing c i rcu i t s run nin g f romthe region between the two of the same color) i t wil l be possibleto co lor the map. Hence we may say tha t a l l o f these opera t ionsare poss ib le on a min imum uncolorab le map. But i t does no tfol low that any map on which al l of these operat ions are possiblei s necessar i ly uncolorab le .Er re ra* descr ibes expl ic i t ly on ly the opera t ion a (or /?) ofthese n ine . Here we wish to s tudy as exhaus t ive ly as poss ib le a l lof the operat ions, and to see i f there is any contradict ion intheir a l l being possible .These n ine opera t ions a re the genera tors o f a g roup , w Thich wewill call the impasse group, and which consis ts of a l l possiblecombina t ions o f t he se ope ra t ions . I n ope ra t ing upon a map bytwo t ransformat ions in success ion , i t i s to be no ted tha t the f i r s topera t ion may move the ver tex of the r ing . Then any second

* Alfred Errera, Du Coloriage des Cartes et de Quelques Questions d'AnalysisSitus, Thesis, Gauthier-Villars, Paris, 1921, 66 pp.


3/7

1935-1 COLORED MAPS 409opera t ion i s re fe r red to th i s ver tex , fo r example , a t r ansposesthe cha in on the le f t o f th i s new ver tex , e tc . Thus , the opera t iona will move the ver tex 144, and hence, i f i t i s performed twice,i t wi l l move the ver tex th rough 288 , e tc . We may note thefo l lowing iden t i t i es :

t2 == Y 2 == 2 = e2 = 02 = i, afi ss fia = t, rj = rj = t.I t fo l lows tha t each of the genera tors has an inverse , and

therefore each member of the impasse group has an inverse .Since we are deal ing with a map of a f ini te number of regions,i t i s ap pa re n t th a t eac h of the opera t ions of the grou p is per iod ic .T h e o p e r a t i o n a has a per iod of a t least twenty, s ince i t can givetwenty different arrangements of the colors of the f ive regionstouching the uncolored reg ion . I t should be no ted tha t in somem a p s a m ay ha ve a per iod of for ty , s ix ty , e tc . T he o pera t io n (3a lso has an apparen t per iod of twenty , whi le f and rj h a v eperiods of fif teen. The operations 7, , e, 6 al l have per iods ofprec ise ly two.We do no t know how many member s o f t he impasse g roupof an uncolorab le map there may be , bu t we can show tha t theremus t be a t l e a s t one hundred twen ty . The re a r e t ha t manypermuta t ions of the co lors o f the f ive reg ions touching the uncolored region, and i t can be shown that each one of these permuta t ions can be e f fec ted by some opera t ion of the impasseg roup . We sha l l no t enumera t e t he ope ra t ions fo r t he va r iouspe rmuta t ions , bu t mere ly a s su re t he r eade r t ha t such ope ra t ionshave been found . I t can be shown tha t a l l o f the permuta t ionsexcept twelve can be ob ta ined by ra i s ing e i ther the pr imi t iveop era t i on s or op era t io ns formed of pa i r s of the gen era to rs to var i ous powers . For these twelve, we can use powers of t r iples . Inal l we can use powers of the fol lowing operat ions only : a , f , a,7a, 7, J*a, af, 777, 77, 77, e/3, 0f7, /32.Whi l e we thus know tha t t he r e mus t be a t l e a s t 120 member so f t he impasse g roup , i t mus t be r emembered tha t t he r e may insome cases be many more . For example , ( /3)8 and (cry)7 eachgives DADBCiox t he r i ng , bu t t he r ema inde r o f t he map migh tconce ivably be co lored d i f fe ren t ly by these opera t ions .E r r e r a p r e sen ted a map on wh ich the ope ra t ion a could berepea ted twenty t imes to re turn to the or ig ina l co lor ing . At each


4/7

410 IRVING KITTELL [June,s tep the co lor ing remained impasse . We wish to po in t ou t a fewaddi t iona l fac t s about Er re ra ' s map, which i s shown in F ig . 2 .T h e o p e r a t i o n s a and f give the fol lowing resul ts shown inFigs . 3 and 4 .

F I G . 2 F I G . 3 F I G . 4W e say tha t t he se ope ra t ion s a r e in th i s pa r t i cu l a r m ap equ iv alent , s ince one color ing can be obtained from the other bys i m p l y p e r m u t i n g ABCD to ADCB t h r o u g h o u t t h e w h o l e m a p .Repet i t ions of f g ive the d iagrams shown in F igs . 5 -7 .

F I G . 5 F I G . 6 F I G . 7N o w n o t e t h a t f4 i s equiva len t to t , s ince th i s map could have

been obta ined f rom the or ig ina l map by ro ta t ing th rough 144and rep lac ing ABCD b y CBDA. I t i s apparen t , therefore , tha t fra ised to any power is equivalent to t , f , f2, or "3.S imi la r ly i t can be shown tha t a2 is not essent ia l ly dif ferentfrom f2, nor a 3 f rom f3 nor a 4 from f4 or t . Hence any power of agives an impasse color ing. I t can also be seen that a 2 0 = t, andth i s i s wha t E r r e r a showed .I t c an also be show n t h a t 0 , 7 , S, 0 , an d rj a re equ iva l en t t of2, 1, t, f3, and f3, respectively; that 7t , Sf, j8f, r tf , and ef areequivalent to f , f , t , t , and f , respect ively; that 0f2, 7^ , S f2, /3f2,a n d rj2 are equiva len t to t , f2, f2, f , and f , respect ively; andt h a t 7 f3, Sf3, 0 r3 , rjf3, and ef3 are equiva len t to f3, f3, f2, f2, andf3, r espec t ive ly .


5/7

1935-1 COLORED MAPS 411H ow ever , th e opera t io n e g ives us the d iag ram shown in F ig .8 . But th is color ing is not impasse , s ince the c i rcui ts do notin te r sec t . The opera t ion 0 f g ives the d iagram shown in F ig . 9 ;

FIG. 8 FIG. 9and th is a lso fa i ls to be impasse s ince the c i rcui ts are bothbroken . I t may be eas i ly shown tha t 0 f2 i s l ikewise ru led out ,as is ef2.In conc lus ion , we say tha t the opera t ions a, j8, 7 , S, *, 77, areposs ib le in any combina t ions o r r a i sed to any power , e tc . ; t h a tthese g ive r ise to only three essent ia l ly d i f ferent color ings ( i fwe n o t e t h a t f3 is me re ly a reflection of f ) ; and th a t e i s possib lew hen 0 i s imposs ib le an d on ly th en , the fo rmer be ing poss ib le ontwo of the three d i f ferent color ings but not on the th i rd .Thus i t i s shown tha t the re i s no poss ib i l i ty o f p rov ing anyc o n t r a d i c t i o n b e t we e n a , /3, 7, , *, a n d 97, n o m a t t e r i n wh a t c o m b i n a t i o n s t h e y a r e u s e d , b e c a u s e i n E r r e r a ' s m a p we h a v e acase in which they a re no t con t rad ic to ry . However , the fo regoing sugges t s tha t the re may be a con t rad ic t ion be tween e , 0 ,and the o the r opera t ions . I f we can f ind some opera t ion o f the

^

A

1/

D

C

B

C

A BD

A

1

D

VlA

/ D

cBCAB

B

! A

71

D

\FIG. 10


6/7

412 IRVING KITTELL [June,impasse g roup o r some se t o f opera t ions such tha t no map wi l lr emain impasse under these opera t ions , then the four -co lo rproblem wil l be solved.The map shown in F ig . 10 i s such tha t each o f the p r imi t iveo p e r a t i o n s ( t h a t is , a , (J, 7 , , e, J", 77, 0) lea ve s t h e co lor in g im passe . Th is shows tha t no con t rad ic t ion can be p roved fo r thep r i m i t i v e o p e r a t i o n s t a k e n s i n g l y .

A

V

1 1B

C |1

BAC

DAB

1 Dk B1 c

71c

kF I G . 11

H ow eve r , in the m ap o f F ig . 10, a 2 resul ts in a color ing whichi s no t impasse . Does th i s mean tha t the re i s a con t rad ic t ion be t we e n e a n d 0 a n d a2? The answer i s nega t ive , a s i s shown bythe m ap o f F ig . 1 1 . H ere e an d 6 are each poss ib le . Also a, a2, a 3 ,and a 4 are each poss ib le , b u t a 5 resu l t s in a non- impas se co lo r ing .

P\~A

B^ C D

^Z D j T c V ^Z ]B j ^ T A J X L B

^\c JL D A T T "^ D C

B

A

c

D

F I G . 12

W e now g ive an o th e r m ap in F ig . 12 , on which th e o pera t ions7 , 8> e a re poss ib le in an y com bina t ions . H ow ever , in th i s m apnone o f the o the r opera t ions ( tha t i s , a, |3, f, rj , 0) are possible.


7/7

1935.] DEFINITIONS OF STIELTJES INTEGRALS 4 1 3But i t shows a t any ra te tha t any e f for t to f ind a cont rad ic t ionbetween 7, 5 , e , or any combinat ion of these, would be fut i le .We have fai led to f ind a map on which al l of the operat ionsof the impasse group a re poss ib le .

N E W Y O R K U N I V E R S I T Y

O N T H E E Q U I V A L E N C E O F T W O M E T H O D SO F D E F I N I N G S T I E L T J E S I N T E G R A L S *B Y B . C . GE T C HE L L

1. Introduction. The S t ie l t j es in tegra l , facj>(x)dg(x)J was or iginally defined for (x) cont inuous on the c losed in te rva l [a, b],a n d g(x) of bounded var ia t ion . The l imi t which g ives r i se to th i sin tegra l i s t aken as the length of the grea tes t sub- in te rva l approaches zero . The above res t r ic t ions on (x) an d g(x), however , are not a t a l l necessary for the exis tence of the l imit , a l though i t fa i l s when the two func t ions have a common poin t o fd i scon t inu i ty . A gene ra l i za t i on wh ich pe rmi t s such d i scon t inu i t ies i s ob ta ined by tak ing the l imi t in the sense of subdiv is ions^to be def ined be low. The Riemann in tegra l i s an ins tance of thef i rs t type of l imit ing process , while the associated Darboux integra l s a re of the subdiv is ion type . These can be shown to be ofthe f i rs t type as wel l . I t i s the purpose of this note to obtaingenera l condi t ions for the equiva lence of the two l imi t s . By thein t roduc t ion of the no t ion of in te rva l func t ions a s imple re s t r ic t ion on the in tegrand i s found to be bo th necessary andsufficient.2. Subdivisions. B y a subdivision, a, of the l inear in te rva lX = [a, b] will be understood a f ini te set of adjacent sub-interva ls whose sum is X. T h e norm of

irving kittell- a group of operations on a partially colored map

Documents