Bojan Mohar- Straight-line representations of maps on the torus and other fiat surfaces

8/3/2019 Bojan Mohar- Straight-line representations of maps on the torus and other fiat surfaces

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DISCRETEMATHEMAT I CS

ELSEVIER Discrete Mathem atics 155 (19 96 ) 173-181

S t r a ig h t - li n e r e p r e s e n t a ti o n s o f m a p s o n t h e to r u s a n d o t h e rf i a t s ur fa cesB o j a n M o h a r 1

University of Ljubljana, Department of Mathematics, Jadranska 19, 61111 Ljubljana, SloveniaReceived l0 J anuary 1993

A b s t r a c tIt is shown that every map on the torus satisfying the obvious necessary conditions has astraight-line representation on the flat torus R 2 / Z 2. The same holds for the Klein bottle, and thetwo-bordered flat surfaces -- the cylinder and the M6bius band.

I . S t r a i g h t- l in e m a p s

B y a we l l - kn ow n r e su l t o f W a gn e r [ 13 ] ( u sua l ly a t t r i bu te d to F f ir y [ 4 ] ) , e ve r y s im p lep la ne g r a p h a dm i t s a s tr a igh t - l ine r e p r e se n ta t ion , i .e . the r e i s a ho m e om or p h i s m o f t hep l a n e s u c h t h a t t h e e d g e s o f t h e g r a p h b e c o m e s t ra i g h t- l in e s e g m e n t s a f t e r p e r f o r m i n gt h i s h o m e o m o r p h i s m . A c o m p a n i o n r e s u l t is t h e t h e o r e m o f S t ei n i tz [ 11 ] t h a t e v e r y3 - c o n n e c t e d p l a n a r g r a p h c a n b e r e p r e s e n t e d a s t h e g r a p h o f a c o n v e x 3 - p o l y t o p e .T h e r e w e r e a t t e m p t s to g e n e r a l i z e S t e i n i t z 's t h e o r e m t o m a p s o n s u r f a ce s o f p o s i t i v ege nus , e . g . [ 5 , 9 ] . B u t i t s e e m s tha t no one t r i e d to e x t e nd the W a gne r - F ~ i r y ' s The -o r e m t o n o n - s i m p l y c o n n e c t e d s u r f a c e s . I n t h i s p a p e r w e f i l l i n t h i s g a p b y p r o v i n ga c o r r e spond ing r e su l t f o r t he tom s , t he K le in bo t t l e , t he c y l inde r , a nd the M 6b iusb a n d . T h e s e a r e t h e o n l y f l a t s u r f a c e s w i t h t h e b o u n d a r y c o m p o n e n t s b e i n g s t r a i g h t .T h e e x i s te n c e o f s tr a ig h t - li n e r e p r e s e n t at i o n s o f m a p s o n t h e t o m s a n d t h e K l e i n b o t tl em i g h t h a v e s o m e a p p l i c a ti o n s i n t h e t h e o r y o f fi l in g s o f th e p l a n e s i n ce t h e i r u n i v er s a lc ove r s g ive r i s e t o p l a ne f i l i ngs . A sho r t d i s c us s ion a bou t t h i s c a n be f ound in [ 7 ,p . 202] .

Ou r p r o o f o f t he e x i s t e nc e o f s t r a igh t - li ne r e p r e se n ta t ion s i s f a i r l y e le m e n ta r y . I tc a n b e e x t e n d e d t o s u r f a c e s o f h ig h e r g e n e r a , a p p l i e d t o t h e ir m o d e l s w i t h c o n s t a n t

1 Supported in par t by the Ministry for Science and Tech nology of Slovenia, Research Project P1-0210-101-92.0012-365X/96/$15.00 1996 Elsevier Science B.V. All righ ts reservedS S D ! 0 0 1 2 - 3 6 5 X ( 9 4 ) 0 0 3 8 1 - 5


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174 B. Mohar/Discrete Mathematics 155 (1996) 173 181

c ur va tu r e - 1 ( t he hype r bo l i c me t r i c ) , a nd w i th t he ge ode s i c s p l a y ing t he r o l e o fs t ra ight - l ine segments . We wi l l not g ive de ta i l s for these cases s ince i t r ecent ly came toour a t t e n t i on t ha t t he e x i s t e nc e o f suc h ge ode s i c r e p r e se n t a t i ons ( f o r su r f a c e s w i thou tb o u n d a r y ) f o l l o w s f r o m t h e C i r c le P a c k i n g T h e o r e m o f K o e b e [ 8 ] , A n d r e e v [ 1 , 2] , a n dT hur s ton [ 12] . T he a dva n t a ge o f ou r p r oo f c om pa r e d t o t he c i r c l e pa c k ing r e su l t s i st ha t i t i s e l e me n ta r y a nd t ha t i t a l so y i e lds a po lynomia l - t ime a lgo r i t hm to p r oduc es t r a igh t - l ine d r a w ings o f g ive n m a ps .

L e t S be a c om pa c t su r f a c e . A m a p on S i s a pa i r M = ( G , S ) , w he r e G i s ac o n n e c t e d g r a ph e m b e d d e d i n S . T o g e t m o r e f r e e d o m , w e d o n o t r e q u i r e th e e m b e d d i n gto be c e l lu l a r , so a ma p c a n ha ve non - s imply c on ne c t e d f a c e s . I f S ha s n on- e m ptybounda r y , a s ~ ~ , t he n w e r e qu i r e f o r e a c h e dge o f G e i t he r t o be d i s j o in t f r om 0S ,ha v ing on ly one o r bo th e ndve r t i c e s on a S , o r e n t i r e ly l y ing on a s . A ma p on t hetorus i s a l so sa id to be a t o r o i d a l m a p . T w o m a p s M = ( G , S ) a nd M I = (GI , S ~ )are e q u i v a l e n t i f t he r e i s a hom e om or ph i sm h : S ~ S t ma pp ing t he g r a ph G o f t hef i rs t ma p i somor p h i c a l l y to t he g r a ph G ~ o f t he s e c ond ma p . I t i s w e l l - know n ( c f .,e . g . , [ 6 ,10 ] ) t ha t tw o ma ps on a n o r i e n t a b l e su r f a c e w i thou t bounda r y a nd w i th a l lf a c e s s imply c onn e c t e d a r e e qu iva l e n t i f a nd on ly i f t he y de t e r m ine t he s a me r o t a t i onsys t e m on t he g r a ph .

A c ompa c t R i e ma nn ia n su r f a c e S ( pos s ib ly w i th bounda r y ) i s f l a t i f e v e r y p o i n tp E S ha s a n e ighbo ur hoo d w hic h i s a f f ine ly d i f f e omor ph i c t o a n ope n se t i n thec lose d upp e r ha l f - p l a ne o f t he E uc l i de a n p l a ne . T h i s i s e qu iva l e n t to t he c ond i t i on tha tt he c u r va tu r e a nd t o r s i on a r e i de n t i c a ll y z e r o , i nc lud ing t he c u r va tu r e o f t he bound-ary aS . T he spec ia l case o f a f la t sur face i s the f l a t t o r u s , t he quo t i e n t spa c e R 2 / Z 2( R 2 / ~ w h e r e ( x , y ) ~ ( x ~ ,y ) m e a n s ( x - x t, y - y~) E Z 2) . A no the r f i a t su r f a c e i sthe f l a t K l e i n b o t t l e . T hi s su r f a c e i s t he quo t i e n t o f t he E u c l i de a n p l a ne R 2 c o r r e -spond ing t o the r e l a ti on ~ g ive n by ( x , y ) ~ ( x + n , ( - 1 ) n y + m ) , n , m E Z . T h ef ia t torus and the f ia t Kle in bot t le a re usua l ly represented as the ident i f ica t ion spaceof t he un i t squa r e by i de n t i f y ing the t op a nd t he bo t t om s ide a nd t he n i de n t i f y ingthe l e f t a nd t he r i gh t , w i th a p r e v ious t u r n by 180 i n c a se o f t he K le in bo t t l e.T he r e a r e tw o a dd i t i ona l f i a t su r f a c e s w i th bounda r y - - t he f l a t c y l i n d e r a nd t hef l a t M f b i u s b a n d . T he y a r e ob t a ine d f r om the un i t squa r e a s w e l l , by i de n t i f y ingon ly one pa i r o f s ide s , t he l e f t a nd t he r i gh t . T o ge t t he c y l i nde r t he y a r e i de n t i -f i e d w i thou t a t u r n , a nd t o ge t t he M 6bius ba nd w e pe r f o r m a tw i s t o f 180 o f ones ide be f o r e t he i de n t i f i c a t i on . I t c a n be show n by us ing t he G a uss - Bonne t f o r mula( c f . [ 3 ] ) t ha t e ve r y c omp a c t f l at su rf a c e i s hom e om or ph i c to one o f t he se f ou r su r -faces .

A s t r a i g h t - l i n e s e g m e n t on a f i at su r f ac e S i s a s e gm e n t o f a ge ode s i c on S . Ama p M on S i s s a id t o be a s t ra igh t - l i ne m ap i f e ac h e d g e o f th e g r a p h o f M i sa s t r a igh t - l i ne s e gme n t . E ve r y s t r a igh t - l i ne ma p M on S i s s im p le , i .e . M has thef o l l ow ing p r ope r t i e s ( s e e F ig. 1 ):

1 . E a c h p a i r o f pa r a l l el e dge s ( e d ge s w i th t he s a me e ndve r t i c e s ) g ive s r i s e t o anonc on t r a c t i b l e c yc l e on S .

2 . N o l oop o f M i s c on t r a c t ib l e .


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0B. M ohar/Discrete M athematics 155 (1996) 173-181 175

C a ) ( b ) ( c )Fig. 1. Forbidden submaps of s imple maps.

3 . I f e i s a l o o p a t t h e v e r t e x v t h e n n o o t h e r l o o p a t v i s h o m o t o p i c t o e ( i. e . , d o e sn o t b o u n d a d i s k t o g e t h e r w i t h e ) .

T h i s i s e a s il y s e e n b y u s i n g t h e G a u s s - B o n n e t f o r m u l a ( c f . [3 ] ). I t i s c l e a r th a tt h e m a p i s s im p l e i f a n d o n l y i f i ts u n i v e r s a l c o v e r h a s n o l o o p s a n d n o p a r a l l e le d g e s .

F o r t h e f l a t t o r u s a n d o t h e r f la t s u r f a c e s w e w i l l a ls o p r o v e t h e c o n v e r s e : A m a p o na f la t s u r f a c e i s e q u i v a l e n t t o a s t r a ig h t - l in e m a p i f a n d o n l y i f it i s s im p l e ( T h e o r e m s3 . 1 , 4 . 1 , 5 . 2 ) .

2 . T r i a n g u l a r ma p s a n d co n tra c ti b l e ed g es

A m a p M = ( G , S ) i s t r i a n g u l a r i f d S C G a n d e v e r y f a c e i s o f s i z e th r e e a n dh o m e o m o r p h i c t o a n o p e n d i sk . T h e s m a l le s t tr ia n g u l a r s i m p l e m a p o n th e t o ru s i ss h o w n i n F ig . 2 . I ts g r a p h c o n s i s ts o f a s i n g le v e r t e x w i t h t h re e l o o p s . W e w i l l s h o wt h a t e v e r y t r i a n g u l a r s i m p l e m a p o n t h e t o r u s c a n b e r e d u c e d t o t h e m a p o f F i g . 2 b ym e a n s o f e d g e c o n t r a c ti o n s .

L e t M b e a t r i a n g u l a r s i m p l e m a p . A n i n t e r io r e d g e e o f M i s c o n t r a c t i b l e i ft h e o p e r a t i o n s h o w n i n F i g. 3 g i v e s r i se to a n o t h e r s i m p l e m a p o n t h e s a m e s u r -f a c e . N o t i c e t h e e d g e s ~ , f l, a n d , r e s p e c t i v e l y , 7 , 6 , o f th e t w o t r i a n g l e s c o n t a i n i n g e ,a r e p a i r w i s e i d e n t i f ie d a f t e r t h e c o n t r a c ti o n . A n e d g e e o n t h e b o u n d a r y o f t h e s u r -f a c e i s c o n t r a c t i b l e i f t h e s i m i l a r o p e r a t i o n a s s h o w n o n F i g . 3 , a d o p t e d t o t h e f a c t

//Fig. 2. The simplest simple triangular map on the toms.


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176 B. M ohar/Discrete M athema tics 155 (1996 ) 173-181\ /

Fig. 3. Edge contraction.

tha t e i s c on ta ine d in on ly one t r i a ngu la r f a c e , g ive s r i s e t o a s im p le m a p on the sa m es u r fa c e . C a l l a n e d g e o f M = ( G , S ) p o t e n t i a l l y c o n t r a c t i b l e i f it i s no t a l oop a nd i ti s e i t he r c on ta ine d in OS , o r one o f i t s e nds i s no t on t3S . I n o the r w or ds , a c on t r a c t iono f a n e d g e w h i c h i s n o t p o t e n t i a ll y c o n t r ac t i b le , w o u l d c h a n g e t h e h o m e o m o r p h i s mtype o f t he su r f a c e . B y de f in i t i on o f a c on t r a c t ion , a n y c on t r a c t ib l e e dge i s po te n t i a l lyc o n t ra c t ib l e . T h e m a p a r i s i n g a f t e r t h e c o n t r a c ti o n o f e i s d e n o t e d b y M / / e .

L e m m a 2 .1 . I f M i s a s im p l e t r i a n g u la r m a p c o n t a i n i n g a t l e a s t o n e p o t e n t i a l l yc o n t r a c t i b l e e d g e t h e n M c o n t a i n s a c o n t r a c t i b l e e d g e .

P r o o f . L e t e b e a p o t e n t i a ll y c o n t r a ct i b l e e d g e o f M a n d s u p p o s e t h a t M / / e i s n o ts im p le . S inc e M c on ta ins no c o n t r a c t ib l e d igons , t he on ly r e a sons f o r t h i s to ha pp e n a r e :

( i ) we ge t a pa i r ~ , f l o f pa r a l l e l e dge s bou nd ing a d i sc , o r( i i ) we ge t a pa i r ~ , f l o f hom otop ic loop s .I n e a c h c a s e ~ a n d f l b o u n d a d i s k i n M / / e . T h i s i m p l i e s t h a t ~ , f l, a n d e b o u n d a

d i sk in M . De n o te by F th i s d i sk ( c f . F ig . 4 f o r t he a c tua l poss ib i l i t i e s i n c a se wh e ne i s a n in t e r io r e dge ) .

A s s u m e n o w t h a t M c o n t a i n s n o c o n t r a c ti b l e e d g e . A m o n g a l l p o t e n t i a ll y c o n t ra c t ib l ee dge s e c hoo se one ( a n d a pa i r ~ , f l ) f o r wh ic h the c o r r e spon d ing d i sk F c on ta ins thes m a l l e s t n u m b e r o f f a c es o f M . S i n c e t h e e d g e s ~ , f l b e c o m e a h o m o t o p i c p a i r o fpa r a l l e l e dge s o r l oops a f t e r t he c on t r a c t ion o f e , t he r e m us t be a t l e a s t one ve r t e x o fG in the in t e rio r o f F . Th i s i s s e e n a s f o l lows . M i s t r i a ngu la r . I n c a se wh e n F i sb o u n d e d b y e x a c t l y th r e e e d g e s ( F i g . 4 ( b ) o r ( d ) ) , F m u s t c o n t a i n a n i n t er i o r v e r t e x ,s inc e F i s no t f a c ia l . I n the r e m a in ing c a se s F i s a 4 - go n ( F ig . 4 ( a ) a n d ( c ) ) . H a v in ga c hor d in F we ge t a c on t r a d ic t ion w i th the m in im a l i ty o f F s inc e the c ho r d c a nr e p la c e ~ o r ft . The r e f o r e , F c on ta ins a n in t e r io r ve r t e x a l so in th i s c a se . Le t f bea n e dge a d ja c e n t t o a ve r t e x in the in t e r io r o f F . The n f i s no t a l oop , a nd by them in im a l i ty o f F , i t f o l lows e a s i ly tha t f i s c on t r a c t ib l e . Th i s c on t r a d ic t s ou r in i t ia la s sum pt ion . [ ]L e m m a 2 . 2 . L e t M b e a s i m p l e t r i a n g u la r m a p o n a f l a t s u r f a c e S , a n d e a c o n -t r a c t ib l e e d g e o f M . l f M ' = M / / e h a s a s t r a i g h t- l i n e r e p r e s e n ta t i o n o n S t h e n t h e


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B. M ohar/Discrete M athema tics 155 (1996) 173-181 177

( a ) ( b )

q

]Fla( c ) ( d )

Fig. 4. Obtaining homotopic loops.

s a m e i s t r u e f o r M . M o r e o v e r , f o r e a c h e > 0 t h e r e is a s tr a i g h t -l i n e r e p r e s e n t a t i o no f M s u c h t h a t th e e d ge l e n gt h s o f M e x c e e d t h e m a x i m a l e d g e l en g th o f M ' b y a tm o s t e .P r o o f . L e t x b e t h e v e r t e x o f M ' c o r r e spo nd ing to e . I t i s e a sy to s e e tha t one c a nr e p la c e x by a ve r y sho r t s t r a igh t - l i ne s e gm e n t r e p r e se n t ing e a nd be ing a b le to d r a wa l l t he e dge s o f M a d ja c e n t t o e u s ing s t r a igh t - l i ne s e gm e n t s . ( To sho w th i s one c a nuse the f a c t t ha t f o r e a c h e dg e f o f M ~ the r e i s a sm a l l e noug h 6 > 0 suc h tha t i nt h e 6 - n e i g h b o u r h o o d n b d ( f , 6 ) : = { p E M [ d i s t ( p , f ) ~ 6 } o f f a n y tw o p o i n t s c a nbe jo in e d b y a s t r a igh t -l i ne s e gm e n t e n t i r e ly ly ing in n b d ( f , 3 ) . ) I f the l e ng th o f e i ssm a l l e r t ha n e the n by the t r i a ngu la r i ne qua l i t y the ne w e dge l e ng ths c a nno t i nc r e a seb y m o r e t h a n e . [ ]

L e m m a 2 . 3 . E v e r y s i m p l e m a p i s c o n t a i n e d i n a t r i a n g u l a r s i m p l e m a p o n t h e s a m es u r f a c e .P r o o f . I t i s w e l l - k n o w n t h a t e v e r y ( s i m p l e ) m a p i s a s u b m a p o f a ( s i m p l e ) m a p o nt h e s a m e s u r f a c e w i t h a l l f a c e s h o m e o m o r p h i c t o a 2 - c e l l a n d s u c h t h a t t h e b o u n d a r yo f th e s u r f a c e is c o v e r e d b y t h e g r a p h o f t h e m a p . G i v e n a m a p M w e m a y t h e r e f o rea s s u m e M h a s t h e s e p r o p e r t i e s .

L e t M b e a s i m p l e m a p . C o n s i d e r a f a c e F o f M . L e t v o, e l , v l , e 2 . . . . . V k - l , e k , V k =V0 b e t h e f a c ia l w a l k o f F , w h e r e e i ( l ~ < i ~ < k ) i s a n e d g e o f M j o i n i n g v e r t i c e s v i - 1


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178 B. Mohar/Discrete Mathematics 155 (1996) 173-181a n d v i on t he boun da ry o f F . Ad d i n F ne w ve r t i c e s v~ . . . . v~ a nd w, a nd j o i n fo ri - - 1 ,2 . . . . . k ve r t i c e s v i an d v~, vi an d V~+l, v an d v~+1 ( i nd i c e s mo du l o k ) , a n d v~a nd w. I t i s e a sy t o s e e t ha t t h i s c a n be done so t ha t t he r e su l t i ng fa c e s r e p l a c i ng Fa re a l l t r i a ngu l a r . The ne w ma p i s s t i l l s i mp l e s i nc e we ha ve no t c ha nge d t he o r i g i na lma p a nd ha ve no t i n t roduc e d l oops o r homot op i c pa ra l l e l e dge s , (P a ra l l e l e dge s ma ya p p e a r o n l y i f a n e d g e e i i s a loop. But in th i s case the para l le l pa i r i s homotopic toe i wh i c h i s non -c on t ra c t i b l e . ) By p e r fo rm i ng th i s ope ra t i on i n e ve ry fa c e o f M we f i nda re qu i re d ma p . [ ]

3 . T h e t o r u sLe t u s a pp l y t he r e su l t s o f the p re v i ous s e c t i on t o t he c a se wh e n t he su r fa c e S i s

t he t oms .

T h e o r e m 3 .1 . A t o r o i d a l m a p i s e q u i v a l e n t t o a s t r a i g h t- l i n e m a p o n t h e f l a t t o r u s i fa n d o n l y i f i t i s s i m p l e .

P ro of . Th e ore m 3 .1 i s a s i mp l e c o ro l l a ry o f Le mm a s 2 .1 -2 .3 a nd t he s t ra i gh t -l i nere p re se n t a t i on o f t he s i mp l e ma p o f F i g . 2 i f we p rove t ha t e ve ry s i mp l e t r i a n -gu l a r t o ro i da l ma p wi t hou t po t e n t i a l l y c on t ra c t i b l e e dge s i s e qu i va l e n t t o t he ma pof F i g . 2 . S i nc e t he t o rus ha s no bound a ry , a ma p w i t hou t po t e n t i a l l y c on t ra c t i b l ee dge s ha s on l y one ve r t e x . By t he Eu l e r ' s fo rmul a we s e e t ha t t he re a re e xa c t l yt h re e l oops ba se d a t t h i s ve r t e x . Now i t i s a n e a sy t a sk t o s e e t ha t t he on l y pos s i -b l e l oc a l ro t a ti on o f t h is g ra ph g i v i ng a s i mp l e t r i a ngu l a r ma p i s t he on e g i ve n i nFig. 2. []

Th e a bove p roo f a c t ua l l y g i ve s a po l ynom i a l - t i me a l go r i thm fo r c ons t ruc t i ng s t ra i gh t-l i ne r e p re se n t a t i ons o f s i mp l e t o ro i da l ma ps .

I t is a s i mp l e c onse qu e nc e o f Le m ma 2 .2 t ha t fo r a n a rb i t r a ry e > 0 , a s i mp l e t o ro i da lma p ha s a s t r a i gh t - l i ne r e p re se n t a t i on whe re e a c h e dge ha s l e ng t h a t mos t x / 2+e . S uc ha re p re se n t a t i on c a n be o b t a i ne d by re qu i r i ng t ha t the l e ng t h o f t he e dge ob t a i ne d bya ver tex spl i t t ing on the k th s tep i s a t m ost e /2 k .

In s t e a d o f be g i nn i ng w i t h F i g . 2 , one c ou l d a s w e l l s t ar t w i t h a n e qu i va l e n t ma p ,fo r e xa mpl e t he one i n F i g . 5 . In suc h a c a se we a l so ge t s t r a i gh t - l i ne r e p re se n -t a t i ons o f s i mp l e t o ro i da l ma ps , un fo r t una t e l y wi t h muc h l e s s ha ndsome f i na l ou t -l ook .

4 . T h e K l e i n b o t t l eThe Kl e i n bo t t l e i s a no t he r su r fa c e wi t h e ve rywhe re f l a t me t r i c . I t s s t a nda rd mode l

i s ob t a i ne d f rom t he un i t squa re i n t he p l a ne b y f i r s t i de n t i fy i ng the t op a nd t he bo t t om


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B. M ohar Discre te Mathemat ics 155 (1996 ) 173-181

Fig. 5. Straightqine map equivalent to the m ap o f Fig. 2.

179

1 / j( a ) ( b )

Fig. 6. The simplest simp le triangular map s on the Klein bottle.

a nd th e n the l e f t a nd the r i gh t s ide , bu t t he se tw o a r e i de n t i f ie d a f t e r a f l ip by 180 o f one o f t he m , so t ha t the l ow e r po in t s on the l e f t a r e i de n t if i e d w i th t he t op po in t son the r i gh t , a nd v i c e ve r sa .T h e o r e m 4 .1 . E v e r y s i m p l e m a p o n t h e K l e i n b o t t l e a d m i t s a n e q u i v a l e n t s t r a i g h t - l i n er e p r e s e n t a t i o n o n t h e f l a t K l e i n b o t tl e .

H a v in g a s im p le t r i a ngu la r m a p on the K le in" bo t t l e , the r e su l t s o f S e c t ion 2 sh owtha t t he re i s a s e que n c e o f e dge c on t r a c t ions l e a d ing to t he m a p w i th a s ing l e ve r t e xa n d t h r e e l o o p s ( s e e th e p r o o f o f T h e o r e m 3 . 1 ) . T h e p r o o f o f T h e o r e m 4 .1 i s t h e n as i m p l e c o n s e q u e n c e o f th e f o l l o w i n g l e m m a .

L e m m a 4 .2 . I f M & a s i m p l e t r ia n g u l a r m a p o n t h e K l e i n b o t t l e w i t h o u t p o t e n t i a l l yc o n t r a c t ib l e e d g e s t h e n M is e q u i v a le n t t o o n e o f t h e s tr a i g h t - li n e m a p s o f F ig . 6 ( a )o r ( b ) .

P r o o f . C o n s i d e r o n e o f t h e l o op s . U p t o h o m e o m o r p h i s m s o f t h e s u r f ac e t h e re a r et h r e e p o s s i b il i ti e s f o r th e h o m o t o p y c l a s s o f th i s l o o p . I t m a y b e s e p a r a ti n g ( n o n -c on t r a c t ib l e ) , 2 - s ide d a nd n on - se pa r a t ing , o r 1 - s ide d . The th r e e t ype s o f t he l oops a r er e p r e se n te d in F ig . 7 a s ~ , f l, 7 , r e spe c t ive ly . I t i s e a sy to s e e t ha t i n c a se w h e n the


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180 B. Mohar~D iscrete M athema tics 155 (199 6) 173-181

; 3a a l 1 7

( a ) ( b )Fig. 7. Simp le closed curves on the K lein bottle.

l oop i s s e pa r a t ing , t he o the r tw o loops a r e 1 - s ide d a nd tha t w e ha ve a m a p e qu iva l e n tto t he m a p in F ig . 6 ( a ) .S u p p o s e n o w t h a t a l l t h r e e l o o p s a r e n o n - s e p a r a ti n g . S i n c e th e g e n u s o f t h e K l e i nbo t t l e i s 2 , t he r e a r e a t m os t tw o ( pa i r w i se ) non - hom otop ic 1 - s ide d loops . S o one o ft h e m i s 2 - s id e d . O n a n y n o n - o r i e n t a b l e t ri a n g u l a te d s u r f ac e w h i c h i s 2 - ce l l d e c o m p o s e dby loo ps , a t l e a s t one o f the l oops i s 1 - s ided . Th a t l oop m us t c r o s s t he f i r s t one . Thes u b m a p c o n s i s t in g o f th e s e t w o l o o p s i s t h u s e q u i v a l e n t t o th e m a p i n F i g . 7 ( b ) . I t h a sone 4 - gon a l f a c e . S inc e ou r m a p i s t ri a ngu la r , t he th i r d l oop jo in s opp os i t e a ng le s o ft h e 4 - g o n . T h e t w o p o s s i b i li t ie s a re i s o m o r p h i c . W e g e t th e m a p o f F i g . 6 (b ) . [ ]

5 . T h e cy li n d er an d th e M fb i u s b an dT o g e t a c o r r e s p o n d i n g r e s u l t f o r t h e r e m a i n i n g f l a t c o m p a c t s u r f a c e s - - t h e c y l i n d e r

a n d t h e M 6 b i u s b a n d - - w e n e e d t o d e t e r m i n e t h e i r m i n i m a l s i m p l e t r i a n g u l a r m a p s .L e m m a 5.1 . T h e o n l y s i m p l e t r ia n g u l a r m a p s o f th e c y l in d e r a n d t h e M 6 b i u s b a n dh a v i n g n o p o t e n t i a l l y c o n t r a c t i b l e e d g e s a r e s h o w n i n F ig . 8 ( a ) a n d ( b ) , r e s p e c t i v e l y .P r o o f . T h e c y l i n d e r h a s t w o b o u n d a r y c o m p o n e n t s . E a c h o f t h e m m u s t h a v e a l o o pon i t. D e no te by v l a n d v2 the tw o ve r t i c e s on the bounda r y . A ny ne w loop a t v i( i = 1 , 2 ) i s e i t he r c on t r a c t ib l e , o r hom otop ic t o t he l oop on the bounda r y . T he r e f o r e ,t h e r e m a i n i n g e d g e s j o i n v l a n d v 2. F o r o n e o f t h e m t h e r e i s o n l y o n e p o s s i b i l i ty ( u pt o e q u i v a l e n c e ) . T h e o t h e r o n e m u s t g o a r o u n d i n o r d e r n o t t o b e h o m o t o p i c t o t h ef i r s t one . The r e su l t i ng m a p i s show n in F ig . 8 ( a ) .

T h e M 6 b i u s b a n d h a s o n e b o u n d a r y c o m p o n e n t , s o w e w i l l h a v e o n l y o n e v e r t e x .T h e r e i s a l o o p o n t h e b o u n d a r y . I t f o l l o w s b y t h e E u l e r ' s f o r m u l a t h a t t h e r e i s e x a c t l yo n e m o r e l o o p . A n y s i m p l e c l o s e d c u r v e o n t h e M 6 b i u s b a n d g o e s t - t i m e s a r o u n d ,w he r e t E { 0 , 1 , 2} . T o ge t a s im p le t r i a ngu la r m a p the o n ly pos s ib i l i t y i s t = 1 w h ic hg i v e s t h e m a p o f F i g . 8 ( b ) . [ ]


9/9

B . M o h a r / D i s c r e t e M a t h e m a t i c s 1 55 ( 1 9 9 6 ) 1 7 3 - 1 8 1 181// l l

( a ) ( b )Fig . 8 . The cy l inder and the M6bius band .

The consequence is our f inal resul t .T h e o r e m 5 . 2 . E v e r y s i m p l e m a p o f th e c y l in d e r o r t h e M r b i u s s t r i p h a s a s tr a i g h t -l i ne r e pre se n ta t ion in the f l a t c y l inde r , o r t he f l a t M db ius s t r ip , r e spe c t i v e ly .

One can use the M rb ius s t rip r ep resen ta tions to ge t s t r a igh t -l ine d rawings o f g raphsem bedded in the p ro jec t ive p lane .

R e f e r e n c e s[1] E.M. Andreev, O n convex polyhed ra in Lobarevski~ spaces, Mat. Sb. (N. S.) 81 (197 0) 445 -47 8;

Engl. t ransl , in Math. USSR Sb. 10 (1970) 413-440.[2] E .M. Andreev , On convex po lyhedra o f fin ite vo lume in Lob arevsk i l space , M at. Sb . (N.S .) 83 (1970)

256-260; Engl. t ransl , in Math. USSR Sb. 12 (1970) 255-259.[3] M.P. do Carmo, Differential G eom etry of Curve s and Surfaces (Prentice-Hall , Eng lewo od Cliffs, N J,1976).[4] 1. Ffiry, On s traight representations of plana r graphs, A cta Sci . Math. Szeged 11 (1948 ) 229 -233 .[5] P . Gr i tzmann , The to ro ida l ana logue to Eberhard ' s Theorem, Mathemat ika 30 (1983) 274-290 .[6] J .L. Gross and T.W. Tucker, Topological Gra ph Theory, (W iley-Interscienc e, New York, 1987).[7] B. Griinbaum, and G.C. Shephard, Til ings and Patterns (Freeman, New York, 1987).[8] P. Koebe, Kon taktproble me au f der konform en Abbildun g, Ber. Verh. Saechs. Akad. Wiss. Leipzig,

Math.-Phys. KI. 88 (1936) 141-164.[9] P. Mani-Levitska, B. Guiga s and W.E. K lee, Rectif iable n-period ic maps, Geom . D edicata 8 (1979 )

127-137.[10] B. Mohar and C. Thomassen, Graphs on surfaces, in preparat ion.[11] E. Steini tz, Polyed er und Raumeinteilungen, Enzykl. Math. Wiss., Part 3AB1 2 (1922 ) 1 -139.[12] W.P. Thursto n, The geometry and topology of 3-manifolds, Princeto n Un iv. Lect. Notes (P rinceton,

N J) .[13] K. Wagner , Bem erkungen zum Vier fa rbenprob lem, Jber . Deutsch . math . Vere in . 46 (193 6) 26 -32 .

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