Coloring Series Parallel Graphs

8/3/2019 Coloring Series Parallel Graphs

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COMBINATORICAAkad~m iai Kiad 5 - Springer-VerlagCOMBINATORICA 0 (4) (1990) 379-392

C O L O U R I N G S E R I E S - P A R A L L E L G R A P H S

P . D . S E Y M O U RReceived M arch 11, 1988Revised Janu ary 20, 1989

We establish a minim ax formula for the chrom atic index o f series-parallel graphs; an d also provethe correctness of a "greedy" algorithm for finding a vertex-colouringof a series-parallel graph.

1 . I n t r o d u c t i o nI n t h i s p a p e r w e p ro v e t w o r e s u lt s , w h i c h a re n o t r e l a t e d t o o n e a n o t h e r e x c e p tt h a t t h e y b o t h i n v o l v e c o l o u r i n g s e r i e s - p a r a l le l g r a p h s . T h e e a s i e r o f t h e t w o c o n c e r n st h e c o r r e c t n e s s o f a " g r e e d y a l g o r i t h m " f o r f i n d i n g a v e r t e x - c o l o u r i n g o f a s e r i e s -p a r a ll e l g r a p h , a n d w e p o s t p o n e a l l f u r t h e r d i s c u s s io n o f t h i s t o S e c t i o n 6. T h e o t h e rr e s u l t c o n c e r n s e d g e - c o l o u r i n g s , a n d w i l l b e d e a l t w i t h i n S e c t i o n s 1 - 5 .A k-edge-co lour ing o f a g r a p h G i s a m a p ~ : E ( G ) ~ { 1 , . . . , k } s u c h t h a t f o rd i s ti n c t e d g e s e , f i f ~ ( e ) = a ( f ) t h e n e a n d f h a v e n o c o m m o n e n d . ( G r a p h s i nt h i s p a p e r a r e f i n i t e , a n d m a y h a v e m u l t i p l e e d g e s b u t n o t l o o p s ; V ( G ) a n d E ( G )d e n o t e t h e v e r te x - a n d e d g e - s et s o f a g r a p h G . ) T h e c h r o m a t ic i n d e x x I ( G ) i s t hem i n i m u m k >_ 0 s u c h t h a t G h a s a k - e d g e -c o l o u ri n g . T h e p r o b l e m o f d e t e r m i n i n g

XII!G) i s N P - h a r d ; i n d e e d , d e c i d i n g w h e t h e r a 3 - c o n n e c t e d c u b i c g r a p h G s a ti s fi e sX ( G ) = 3 o r n o t i s N P - c o m p l e t e [ 6 ] . O n t h e o t h e r h a n d , f o r p l a n a r g r a p h s G i ti s n o t k n o w n w h e t h e r d e t e r m i n i n g x I ( G ) i s N P - h a r d ; a n d i n d e e d f o r p l a n a r g r a p h st h e re i s a c o n j e c t u r e d m i n i m a x f o r m u l a ( w h i c h , if t r u e , w o u l d b e c o n v e r t ib l e i n t o ap o l y n o m i a l a l g o r i t h m , v i a [ 3 ,4 ] ) w h i c h w e n o w d i sc u s s .L e t A ( G ) d e n o t e t h e m a x i m u m v a l e n c y o f t h e v e r ti ce s o f G ( t h e va l en cy o f av e r t e x i s t h e n u m b e r o f e d g e s i n c i d e n t w i t h i t ). T h e n c l e a r ly x t (G ) >_ A (G ) , b u te q u a l i t y n e e d n o t o c c u r ( f o r e x a m p l e , w h e n G = K s ) . T h e r e i s, h o w e v e r , a n o t h e r

u s e f u l l o w e r b o u n d o n Xt(G) , as fo l lows . For X C_ V (G ), w e d e n o t e b y ~ " t h e s e to f e d g es o f G w i t h b o t h e n d s i n X . I n a n y x l ( G ) - e d g e - c o lo u r i n g ~ o f G , a t m o s t[ ] X I / 2 j e d g e s i n X h a v e t h e s a m e c o l o u r ( w h e r e ~pJ d e n o t e s t h e g r e a t e s t i n t e g e r nw i th n _< p) , an d h enc e IXI _< x I ( G ) . [ I X I / 2 J . T h i s f ol lo w s f r o m o u r o t h e r b o u n d

AM S subjec t classification (1980): 05 C 15


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380 P.D. SEYMOUR

xI (G ) >_ A (G ) i f I X I is e v e n , b u t i f I X I i s o d d ( a n d a t l e a s t 3 ) i t s o m e t i m e s p r o v i d e sn e w i n fo r m a t io n . L e t( 2 i x I )F ( G ) = m a x - - 1 : X c v ( a ) , i X l _> 3 a n d o d d .

W e h av e s h ow n t h e n t h a t x ' (G) >_m a x ( A ( G ) , I F ( G ) ] ) ( w h e r e [p ] m e a n s - [ - p J )f o r e v e r y g r a p h G . I t w a s c o n j e c t u r e d i n [ 1 0 ] ( a n d r e m a i n s o p e n ) t h a t( 1 . 1 ) ( C o n j e c t u r e ) . I f G i s p la n a r th en x ' (G ) = m a x ( A ( G ) , I F ( G ) ] ) .

E q u a l i t y d o e s n o t h o l d f o r g r a p h s i n g en e r al ; f o r in s t a n c e i f G i s t h e P e t e r s e ng r a p h , o r t h a t g r a p h m i n u s o n e v e r t e x , t h e n x I ( G ) - - 4 b u t A ( G ) = F ( G ) = 3 .C o n j e c t u r e ( 1 .1 ) is n o d o u b t d i f fi c u lt , if t r u e , b e c a u s e i t c o n t a i n s t h e f o u r - c o l o u rp r o b l e m ( fo r i t i m p l ie s t h a t 3 - c o n n e c t e d c u b i c p l a n a r g r a p h s a r e 3 - e d g e -c o l o u r a b le ) .O u r o b j e c t h e r e i s t o p r o v e C o n j e c t u r e ( 1 .1 ) f0 ~ s e r i e s - p a r a l l e t g r a p h s . ( P l e a s e n o t et h a t f o r s imple s e r i e s - p a r a l l e l g r a p h s t h i s i s n o t v e r y d if f ic u l t; b u t w e w i s h t o p r o v ei t p e r m i t t i n g m u l t ip l e e d g e s. ) M a r c o t t e [ 7] s h o w e d t h a t xr(G) _ 0 b e s o m e f i x e d i n t e g e r. I f G i s a g r a p h a n dX c_ V (G ), w e d e f in e 5 ( X ) = ~ G ( X ) = k l X I - 2 1 ~ t ~ -W e s a y t h a t X i s odd i f IXI i so d d , a n d even o t h e r w i s e .( 1 . 3 ) F( G ) __ k f o r ever y odd X C_ V (G ).P r o o f . L e t X C V ( G ) b e o d d . T h e n 2 I X ] _< k ( I X I ~ - 1 ) i f a n d o n l y i f k lX I - 5 (X ) _ k. S i n c e 5 ( X ) = k i f I X I = 1 , t h e r e s u l t f o l lo w s . |

W e s h a l l p r o v e( 1 . 4 ) Le t G be a series-parallel graph with ,A(G) _ k fo r everyodd X c_ V (G ). Th en x~(G)


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C O L O U R I N G S E R I E S - P A RA L L E L G R A P H S 381

2 . C o l o u r i n g r o o t e d g r a p h sW e d e n o t e t h e v a l e n c y o f a v e r t e x v o f a g r a p h G b y d ( v ) o r d a ( v ) w h e n t h e r eis a d a n g e r o f a m b ig u i t y . A g r a p h G i s f r e e i f A ( G ) , F ( G ) _< k ; t h a t i s, b y ( 1 . 3 ), i f

(i ) d ( v ) < k f o r e a c h v E V ( G ) a n d(ii) 5 ( X ) > k f o r e a c h o d d X C_ V (G ) .A rooted graph i s a t r i p l e ( G , s , t ) , w h e r e G i s a g r a p h a n d s , t E V ( G ) a r e d i s t i n c t. W edef ine p(G , s , t ) (o r p ( G ) , w h e n t h e r e i s n o d a n g e r o f a m b i g u i t y ) t o b e t h e m i n i m u mv a lu e o f ~ f ( X ) / 2 t a k e n o v e r a l l e v e n X C_ V ( G ) w i th s , t e X . ( T h i s is w e l l - d e f in e d ,s i nc e { s, t } i s e v e n . ) S im i l a r l y w e d e f in e q(G , s , t) (o r q ( G ) ) t o b e t h e m i n i m u m o f( ~ ( X ) - k ) / 2 , t a k e n o v e r a ll o d d X C_ V (G ) w i t h s , t e X , i f s u c h a n X e x i s ts , a n do th e r w i s e q(G , s , t ) = k .(2.1) p(G ) , q (G ) a r e in t e ge r s , and i f G i s f r e e t he n p (G ) , q (G ) _ 2k - d (s ) - d ( t ) .P r o o f . T h e r e s u l t h o ld s if I V ( G ) [ = 2 , a n d w e a s s u m e th a t I V ( G ) I _> 3 . Ch o o s eX , Y C_ V ( G ) w i t h s , t E X , Y a n d w i t h X e v e n a n d Y o d d , s u c h t h a t p ( G ) = 5 ( X ) / 2a n d q ( G ) = ( 5 ( Y ) - k ) / 2 . T h u s

5 ( X ) + 5 ( Y ) = 2 p ( G ) + 2 q ( G ) + k .L e t r b e t h e n u m b e r o f e d g e s o f G w i t h o n e e n d i n X n Y a n d t h e o t h e r i n X A Y .T h e n

2 5 ( X n Y ) + 5 ( X A Y ) - 2 r < 5 ( X ) + 5 ( Y ) .M o r e o v e r , 5 ( X A Y ) > k s in c e X A Y i s o d d a n d G i s f re e , a n d i t f ol lo w s t h a t

5 ( X n Y ) - r


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3 8 2 P . D . S E Y M O U R(2 .4 ) I f G is f ree the n q(G ) >_ k - d( s) - d( t ) an d q(G ) >_ O.P r o o f . T h e f i r s t f o ll o w s f r o m ( 2 . 1 ) a n d ( 2 .3 ) . F o r t h e s e c o n d , l e t X C_ V (G ) b e o d d ,w i t h s , t E X . T h e n b ( X ) >_ k s in c e G i s f r e e , a n d s o ( ~ ( X ) - k ) / 2 >_ O. T h e r e s u l tfo l lows . |

A goal f o r a r o o t e d g r a p h ( G , s , t ) i s a p a i r ( S , T ) o f s u b s e t s o f { 1 , . . . , k } . I t i sa t t a i ne d b y ( G , s , t ) i f t h e r e i s a k - e d g e - c o l o u r i n g ~ o f G s u c h t h a t f o r e a c h e E E ( G ) ,i f e ,-~ s th en K . (e ) r S , a nd i f e ,,~ t the n a ( e ) ~ T . [We w r i te e ~ s i f e is inc ide n t w i ths.] W e s a y t h a t a g o a l ( S , T ) is reasonable i f I S U T [ < p ( G , s, f ) , [S N T I _ 0 w h e r et h e s u m i s t a k e n o v e r al l e d g e s f r o m X t o Y . B y e n u m e r a t i n g a ll s u c h p a r t it i o n s ,t h e r e s u l t f o ll o w s . |( 3 .2 ) L e t a l , a 2, b l ,b 2 , c l , c 2 , d , e , f b e i n te g e rs . T h e r e a r e i n te g e r s X l , X 2 , y z , y 2 , z > Osuch that (x l , x2 , y l , y2, z ) A < (a l , a2 , b l , b~, e l , c2 , d , e , f ) , where


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COLOURING SERIES-PARALLEL GRAPHS 383

a 0

0 d

b 0

Fig. 1

i 0 - 1 0 1 0 1 0 - 10 1 0 - 1 0 1 1 0 - 1A = 0 0 - 1 0 1 0 0 1 - 1 ,0 0 - 1 0 1 0 1 - 10 - 1 - 1 1 1 1 1 - 1

i f and only i f all the fol lowin g 18 qua nti t ies are non-negative: al , a2, c l , c2, d, e ,bl + c t , b 2 +c 2, al + b t + e , a 2 + b2 + e , bl + d + e , b2 + d + e , cl + c 2 + f , d + e + f ,a t + a 2 + e + f , a t + c 2 + e + f , a 2 + c l + e + f , b t + b2 + d + e .T h e p r o o f is a n a l o g o u s to t h a t f o r ( 3 .1 ) , e x c e p t t h a t t h e r e i s n o g r a p h c o rr e -s p o n d i n g t o t h a t o f F i g u r e 1, a n d s o w e n e e d a v e r si o n o f H o f f m a n ' s t h e o r e m f o rr e g u l a r m a t r o i d s [ 9] . W e o m i t t h e d e t a i ls . I n c i d e n t a l ly , t h e m e t h o d o f s i m p l y e l im i -

h a t i n g v a r i a b l e s o n e b y o n e a l s o p r o v id e s a r e a s o n a b l e a l t e r n a t i v e p r o o f o f (3 .2 ); w i t h( 3 . 1 ) , t h i s m e t h o d s e e m s t o b e t o o c u m b e r s o m e f o r c o n v e n i e n t h a n d c a l c u l a t i o n .4 . P i e c i n g t o g e t h e r s u c c e s s f u l g r a p h s

A separat ion o f a g r a p h G i s a p a i r ( G 1 , G 2 ) o f s u b g r a p h s o f G w i t h n o c o m m o ne d g e s a n d w i t h G 1 U G ~ -- G .(4 ,1) Le t G be free, le t s l , s2 E V (G ) be dist inct , and let ( G I , G2) be a separation ofG w i t h s t e V ( G t ) , s2 e V ( G 2 ) a n d V ( G t n G2) = {t} where t ~ st, s2. I f ( G t , s t , t )and (G2 , s2, t) are both successful the n so is (G, s l , s2 ) .P r o o f . L e t ( S t , $ 2) b e a r e a s o n a b l e g o a l f o r ( G , s t , s : ) ; w e s h a l l s h o w t h a t i t i sa t t a i n e d . W e a b b r e v i a t e p ( G t , s t , t ) b y p (G1 ) , e tc . F o r i = 1 , 2 le t di( t ) b e t h ev a l e n c y o f t i n G i.


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384 P.D . SEYMOUR( 1 ) I f t h e r e a r e i n t e g e r s x , y , z , w > 0 s u c h t h a t

x < _ I S , - S ~ ly


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COLOURINGSERIES-PARALLELGRAPHS 385F r o m ( 3 .1 ) , ( 2 .1 ) a n d ( 1 ), t o c o m p l e t e t h e p r o o f i t s u ff ic e s t o c h e c k t h a t t h ef o l lo w i n g 2 6 i n e q u a l i t i e s h o l d :

( a ) I S , - S ~ l > 0( b ) k - I S , u S = l > _ o( c ) I s ~ - s , I _> o( d ) I s , n s = l >_ o( e ) k - d , ( t ) - d = ( t ) >_ 0( f ) q ( C , ) >_ 0( g ) p ( C ~ ) - I S = l _ > 0( h ) p ( G , ) - I S , I > _ 0( i ) q ( G z ) >_ 0( j ) q ( G , ) + q ( G 2 ) - ] $1 N $ 21 >_ 0( k ) q(a~ ) + p(a= ) - IS , U & l >- 0( l) p ( C ~) + p ( G 2) - I S , n S2 1 - k >_ 0( m ) p ( C , ) + q ( a : ) - I S , u S = l > _ o( n ) k - d l ( t) ~ 0( o ) k - d2 (t) >_ 0( p ) q ( G 2 ) + k - I S 2 1 - d ~ ( t ) >_ 0( q ) q ( G ~ ) + k - I S , I - d 2 ( t ) >_ 0( r ) p ( G ~ ) - d l ( t ) >_ 0( 8 ) p ( C , ) - d 2 ( t ) >_ 0( t ) p ( G 2 ) + q ( G 2 ) - I S 2 I - d , ( t ) > 0( u ) p ( G , ) + q (G ~ ) - I s , J - d 2 ( t ) > 0( v ) p ( G , ) + q ( G 1 ) + q ( G = ) - I S , l - l S = l > _ _ o( w ) q ( C , ) + p ( G = ) + q ( a = ) - I S , I - l S = l _ > o( x ) p ( a l ) + q ( G 1 ) + p ( G 2 ) - 1811 - k _> 0( y ) p ( G , ) + p ( G = ) + q ( C ~ ) - IS~ l - k >_ 0( z ) p ( a x ) + q ( G , ) + p ( G 2 ) + q ( G ~ ) - I S , I - [$ 21 - k >_ 0

N o w i n e q u a l i t i e s ( a ) , ( b ) , ( c ), ( d ) a r e tr i v i a l , a n d ( e ), ( n ) , ( o ) h o l d b e c a u s ed (t ) _ I S , l , d = ( t ) , I S = I + d ( s = ) + d ~ ( t ) - k ,q ( G x ) > _ O , I S , I + d = ( t ) - k , I S , I + I S = I + d ( s 2 ) + d 2 ( t ) - 2 k .

B u t s i n c e d ( s , ) + ] S i I ~ k , d ( s 2 ) + I S 2 1 _ < k a n d d , ( t ) + d 2 ( t ) __ k - d ( s x ) _> I S , hp ( C , ) >_ k - d , ( t ) >_ d = ( t ) >_ d 2 ( t ) + I S = I + d ( 8 2 ) - k ,

a n d f r o m ( 2 . 4 ) t h a t q ( G , ) >_ 0 a n dq ( G ~ ) > k - d , ( t ) - d ( s , ) > d2 ( t ) + IS , I - k >_ [S , [ + [$21 + d ( s 2 ) + d 2 ( t ) - 2 k .


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3 8 6 P . D . S E Y M O U R

T his p roves ( f ) , (h ) , (q ) , ( s ) , (w) , (y ) , an d (g ) , ( i ) , (p ) , ( r ) , (v ) , (x ) fo l low bys y m m e t r y . F o r ( t ) , ( u ) a n d ( z) i t su f fi ce s b y ( 2 .3 ) t o s h o w t h a t2k - d (s2 ) - d l ( t ) - d2(t) - [Se[ > 0,2 k - d ( s l ) - d l ( t ) - 6 ( t ) - [$1[ >_ O,3 k - - d ( s l ) - d ( s 2 ) - d l ( t ) - d2(t) - [$1[ - - I S 2 [ _ ~ 0

a n d t h e s e f o l lo w si n c e d ( 8 i ) + I S i l < k ( i = 1 , 2) an d d ~ ( t ) + d e ( t ) < k . F i n a l l y ,fo r in e q u a l i t i e s ( j ) , ( k ) , ( 1 ) , (m ) , c h o o s e P i , Q i c_ V ( G i ) w i t h s i , t E P i , Q i a n dw i t h P i e v e n a n d Q i o d d s u c h t h a t p ( G i ) = 5 ( P i ) / 2 a n d q ( G i ) = ( 5 ( Q i ) - k ) / 2 .( T h e f o ll o w i n g p r o o f m a y e a s i ly b e a d a p t e d t o h a n d l e t h e c a s e s w h e r e [V (G 1)] = 2o r IV(G=)[ = 2 , a n d we o m i t t h o s e d e t a i l s . ) T h e n P ~ U P 2 , Q~ u Q2 a r e o d d , a n dP1 U Q2 , P2 u Q1 a re even , an d so

~ 5 ( P ~ U Q 2 ) , 2 5 ( P 2 u Q 1 ) >_ p ( G , s ~ , s 2 ) >_ I S , u $ 2}2 ( 5 ( P ~ U P 2 ) - k ) , 2 ( 5 ( Q ~ u Q 2 ) - k ) > q ( G , s l , s 2 ) > IS 1 n S e ].

B u t 5 ( P ~ U P ~ ) = 5 ( 1 ) 1 ) + 5 ( P ~ ) - k a n d s op ( G ~ ) + p ( G 2 ) = ~ 5 ( P 1 ) + 1 5 ( P 2 ) = I (5(P~ U P2) + k) > lS~ n S el + k .

Th i s p ro v e s (1) a n d ( j ), ( k ) , (m ) fo ll o w s i m i l a r l y . Th i s c o m p l e t e s t h e p ro o f . |(4 .2) L e t G b e f r e e , l e t s , t E V ( G ) b e d i s t i n c t , a n d l e t ( G 1 , G 2 ) b e a s e p a r a t i o n o fG w i t h V ( G 1 f 3 G 1 ) = { s , t } . / f ( G 1 , s , t ) a n d ( G 2 , s , t ) a r e b o t h s u c c e s s fu l t h e n s o i s( G , s , t ) .P r o o f . L e t ( S , T ) b e a r e a s o n a b l e g o a l fo r ( G , s , t ) . L e t d i ( s ) , d i ( t ) b e t h e v a l e n c i e sof s , t in G i ( i = 1 , 2 ) . A g a i n w e a b b r e v i a t e p ( G l , S , t ) b y p ( G 1 ) , e t c .( 1 ) I ] t h e r e a r e i n t e g e r s x l , x 2 , Y l , Y 2 , z > 0 s u c h t h a t

x l _< I S - T I ,z = < I T - S [ ,

- - 2 1 - - Y l - - Z


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COLOURINGS E R I E S - P A R A L L E L G R A P H S 387

] Z I = z + k - p ( G 2 ) . T h i s i s p o s s i b l e s i n c e t h e s e c a r d i n a l i t i e s a r e a l l n o n - n e g a t i v e(for p ( G ~ ) < k b y ( 2 . 1 ) ) a n dY l + Y2 - F z

b y (2 . 1 ) . W e d e f i n e+ k - p ( G 2 ) _ 0p ( G 2 ) - I T [ - d , ( t ) >_ 0p ( G ~ ) - [ S l - d 1 ( 8 ) > _ 0q ( G , ) + p ( G 2 ) - k - [ S D T [ >_ 0p ( G , ) + p ( G 2 ) - k - I S U T I _> 0k - I T I - d l ( t ) - d 2 ( t ) _> 0k - [ 5 , [ - - d l ( 8 ) - e 2 ( 8 ) ~ op ( G 1 ) - d :( t) - IT[ _> 0p( G 1) - d2(8 ) - ISI _> 0p (G1 ) + q (G1 ) + p (G2 ) - k - d 2 ( t ) - I S I - I T I >_ 0p ( G 1 ) + q ( G , ) + p (G2 ) - k - d 2(8) - I S I - I T I _> 0p(G 2) q - q (G2) - d l ( s ) - d l ( t) - 15,] - ITI -> 0


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3 8 8 P . D . S E Y M O U R .

( n ) p ( G ~ ) + q ( G ~ ) + p ( G 2 ) + q ( G 2 ) - 2 k - 1 8 1 - I T I _> 0( o ) p ( G ~ ) + q ( G 2 ) - k - I S n T { >_ o( p ) p ( G ~ ) + p ( G 2 ) + q ( G 2 ) - k - d r ( s ) - IS l - ITI > 0( q ) p ( G ~ ) + p ( G 2 ) + q ( G 2 ) - k - d ~ ( t ) - [S[ - IT[ >_ 0( r ) p ( G t ) + q ( G t ) - d ~ ( s ) - d 2 ( t ) - IS I - IT I _ > 0

N o w i ne qu a l i t i e s ( a ) , ( b ) a r e t r i v i a l , a n d (g ) , ( h ) ho l d s inc e (S , T ) i s a r e a s ona b l egoa l f o r (G , s , t ) . B y (2 . 2 ) , p ( G I ) > k-dl(8) _> d 2 ( s ) + [ S I ( s ince d ( s ) + [ S I < k ) a n d so( j) ho l ds . B y t h e s y m m e t r y , ( c ), ( d ) a nd ( i) ho l d . B y (2 . 3 ) , s inc e p (G 1) _> d2 ( s ) + [S [a n d d l ( t ) + d2( t ) + IT[ < k , i t fo l lows th a tp ( G 1 ) + p ( G 2 ) + q ( G 2 ) >_ 2 k - d 2 ( t ) + [ S[ >_ k + d l( t ) + IS] + IT[

a n d s o ( q ) h o l d s; a n d ( k ) , ( 1) , ( p ) f o ll o w f r o m t h e s y m m e t r y . F o r ( m ) , ( n ) , a n d ( r) i ts u f f ic e s by (2 .3 ) t o s h ow t h a t2 k - d l ( s ) - d l ( t ) - d2(s ) - d 2 ( t ) - ]S I - ITI _ 0,

w h i c h i s t r u e s in c e d l ( s ) + d 2 ( s ) + [S[ _< k and d l ( t ) + d 2 ( t ) + [TI _ IS[ , ands o (o ) ho l ds i f ]V (G 2)] = 2 , a n d s i m i l a r l y ( e) ho l ds i f I V ( G 1 ) ] = 2 .) T h en P1 U P2 ise v e n , a n d / ) 1 U Q 2 , Q1 u P 2 a r e o d d , a n d s oI S U T I < p ( C , s , t ) < 1 5 ( p ~ u p 2 ) = 1 ( 5 ( P I ) + 5 ( P = ) - 2 k ) = p ( G , ) + p ( G 2 ) - kI S n T I


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COLOURING SERIES-PARALLEL GRAPHS 389

(1 ) I f ( F , F t) E :~ t h e n b o th F a n d F t h a v e p a t h s j o i n i n g s a n d t , w h e r eV ( F n F ') = {s , t} .L e t ( F , F ' ) E 5 ~ , w i th V ( F N F ' ) = {s , t } . W e s h a ll p r o v e b y i n d u c t i o n o n I E ( F ) [t h a t ( F , s , t ) i s s u c c e s s fu l . B y ( 5 .1 ) w e m a y a s s u m e t h a t [ E ( F ) I > 1.(2 ) I f t h e r e i s a s e p a r a t i o n ( F 1 , F ~ ) o f F w i t h V ( F 1 n F 2) = { s , t } a n d E ( F 1 ) ,E(F 2) ~ O then (F , s , t ) i s success fu l .F or (F1 , F2 t3 F t) , (F2 , F1 t_)F ' ) e Y , a n d s o f r o m o u r i n d u c t i v e h y p o t h e s i s ( F 1 , s , t )a n d ( F ~ , s , t ) a r e su c c e s s f u l. S i n c e F i s f r e e w e d e d u c e f r o m ( 4 .2 ) t h a t ( F , s , t ) iss u c c e ss f u l a s r e q u i r e d .(3 ) I f t h er e d o n o t e x i s t tw o p a t h s o f F b e tw e e n s , t w i t h n o c o m m o n v e r ti c esexcep t s , t t hen (F , s , t ) i s success fu l .F o r b y M e n g e r ' s t h e o r e m t h e r e w o u l d b e a s e p a r a t i o n ( F1 , F 2 ) o f F w i t h s EV ( F 1 ) , t e V ( F 2 ) a n d V ( F 1 A F 2 ) = { u } w h e r e u ~ s , t . T h e n E ( F I ) , E ( F 2 ) ~ Oa n d F 1 , F ~ a r e f r e e , a n d s o ( F 1 , s , u ) a n d ( F 2 , t , u ) a r e s u c c e s s f u l , f r o m o u r i n d u c t i v eh y p o t h e s i s ; a n d h e n c e s o i s ( F , s , t ) , f r o m ( 4 . 1 ) .T o c o n c l u d e , w e p ro v e t h a t o n e o f ( 2 ), ( 3 ) a p p li e s. F o r su p p o s e n o t . T h e n s , ta r e n o t a d j a c e n t i n F , b e c a u s e i f t h e y w e r e t h e n s i n c e [ E ( F ) [ > 1 t h e r e w o u l d b e as e p a r a t i o n a s i n ( 2 ). S i n c e (3 ) d o e s n o t a p p l y , t h e r e a r e p a t h s P 1 , P 2 o f F b e t w e e ns , t w i t h n o c o m m o n v e r t e x e x c e p t s , t. S i n c e s , t a r e n o t a d j a c e n t i n F , P~ a n d P ~b o t h h a v e i n t e r n a l v e r t ic e s . S i n c e t h e r e i s n o s e p a r a t i o n a s i n ( 2) , t h e r e i s a p a t h/)3 o f F f ro m V ( P 1 ) t o V ( P 2 ) w i t h s , t V ( P a ) , a n d w i t h n o v e r t e x in V ( P I (3 P 2 )e x c e p t i t s e n d s . L e t P 4 b e a p a t h o f F I b e t w e e n s a n d t . T h e n P 1 (J P 2 U P 3 U P 4 i s as u b d i v is i o n o f K 4 , a c o n t r a d i c t i o n . T h i s c o m p l e t e s t h e i n d u c t i v e p r o o f t h a t ( F , s , t )i s s u c c e s s f u l , f o r e v e r y ( F , F I ) E 5 .T o c o m p l e t e t h e p r o o f o f t h e t h e o r e m , l e t f E E ( G ) w i t h e n d s s, t , l e t F = G \ f ,a n d l e t F ' b e t h e g r a p h w i t h V ( F ' ) = { s , t } , E ( F ~) = { f } . N o w ( F , F ' ) E ~ s in c e[ E ( G ) I > 2 , a n d s o ( F , s , t ) i s s u c c e s s fu l . L e t S = T = { 1 } . N o w i f X C V ( G ) w i t hs , t E X th e n 5 F ( X ) = 5 G ( X ) + 2 ; a n d s o i f X i s e v e n t h e n 5 F ( X ) _> 2 , a n d i f Xi s o d d t h e n 5 F (X ) >__ 5 G (X ) + 2 _> k + 2 s ince G is f ree . H en ce p ( F , s, t , ) >_ 1 andq ( F , s , t ) >_ 1 . M or eo ve r

dF (S) + IS I = d F( S) + 1 = dG (s )


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r e c u r s iv c l y s t r i p p i n g a w a y v e r t ic e s w i t h a t m o s t t w o n e i g h b o u rs . T h e r e is a n o t h e rs i m p l e a l g o r i t h m w h i c h w o r k s f o r s e r i e s - p a r a l l e l g r a p h s a s w e s h a l l s e e , b u t c u r i o u s l yi t s c o r r e c t n e s s i s m u c h l e s s o b v i o u s .H e r e i s a n a l g o r i t h m w h i c h a t t e m p t s t o f in d a k - c o l o u r in g o f a g e n e r a l i n p u tg r a p h G .A l g o r i t h m . W e re c u r s i v e ly c o l o u r v e r ti c e s, o n e a t e a c h i t e r a t i o n . T h u s , a t t h eb e g i n n i n g o f t h e f i r s t i t e r a t i o n ~ ( v ) i s n o t d e f i n e d fo r a n y v e r t e x v . A t t h e b e g i n n i n go f t h e i t h i t e r a t i o n , w e d e n o t e b y X i t h e s e t o f a l l v e r t ic e s v f o r w h i c h ~ ( v ) i s d e f i n e d ;t h u s IXi l = i - 1. F or v 9 V ( G ) w e d e f i n e

N i ( v ) = { t ~ i u ) : u 9 X i i s a d j a c e n t t o v } .T h e i t h i t e r a t i o n p r o c e e d s a s f ol lo w s . I f X i = V ( G ) w e a n n o u n c e " su c c es s " a n d s t o p .O t h e r w i s e w e c h o o s e v 9 V ( G ) - X i w i t h IN i ( v ) [ m a x i m u m . I f N i ( v ) = { 1, 2 , . . . , k }w e a n n o u n c e " s t u c k " a n d s t o p . O t h e r w i s e w e c h o o s e j w i t h I < j _< k a n d j q~ N i( v ) ,d e f i n e t~ (v ) = j , a n d r e t u r n f o r t h e n e x t i t e r a t i o n .I f k - - 2 a n d G i s b i p a r t i t e t h e n t h e a l g o r i t h m a l w a y s su c c e e d s , b u t i n g en e r a li t is n o t m u c h g o o d . F o r i n s t a n c e , i f k >_ 3 t h e a l g o r i t h m c a n g e t s t u c k e v e n i f G i sb i p a r t i t e , a s t h e s e c o n d e x a m p l e ( d u e t o N . A l o n ) i n F i g u r e 2 s h o w s .

3

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1

2

?Fig. 2

W i t h k = 3 , o n e i m p l e m e n t a t i o n o f t h e a l g o r i t h m o n e i t h e r g r a p h f r o m t h e F i g u r ec h o o s e s v e r t i c e s i n v e r t i c a l o r d e r , s t a r t i n g f r o m t h e t o p , a n d d e f i n e s t~ i t e r a t i v e l y a ss h o w n , a n d g e t s s t u c k a t t h e b o t t o m v e r t e x ; a n d y e t b o t h g r a p h s a r e 3 - c o l o u r a b l e .O n t h e o t h e r h a n d , w e sh a ll s ho w( 6 .1 ) I f G i s s e r i e s -p a r a l l e l a n d k -co l o u r a b le t h en t h e a l g o r i t h m s u cceeds .

I n c i d e n ta l l y , t o s e e t h e m o t i v a t i o n f o r t h i s r e su l t , i t m a y h e l p t o v i e w i t a sa s s e r t in g n o t t h a t " h e r e i s y e t a n o t h e r a l g o r i t h m t o c o lo u r s e r i e s - p a r a l le l g r a p h s " , b u t


13/14

C O L O U R I N G S E R I ES - PA R A L L E L G R A P H S 3 9 1r a t h e r t h a t " h e re is a s u r p r i s i n g l y r ic h c l as s o f g r a p h s f o r w h i c h o u r n a i v e a l g o r i t h mw o r k s " .

S i n c e t h e a l g o r i t h m s u c c e e d s if k < 2 a n d G i s k - c o l o u r a b le , w e m a y a s s u m e t h a tk >__ 3 . T hu s (6 .1 ) i s im p l i e d by t he fo l low ing .(6 .2) Le t n > O , l e t V l , . . . , v t b e d i s t i n c t v e r t ic e s o f a s e r i e s - p a r a l l e l g r a p h G , a n d l e tto : { V l , . . . , v t - 1 } "-* { 1 , . . . , n } b e s u c h t h a t f o r 1 < i < t, [ N i ( v l ) l >_ I N i ( v ) [ f o r e a c hU e V ( e ) - { V l , . . . , ~ ] i - i } , w h e , ~

N i ( v ) = {a (V h) : 1 2 . S o m e v e r t e x v e V ( G ) - X i s a t i s f i e s N i ( v ) r qJ si nc e G isc o n n e c t e d , X i ~ 0 a n d X i # V ( G ) ; a n d s o b y h y p o t h es i s , [ N i ( v i ) } > } N i ( v ) [ > O .T h u s v i i s a d j a c e n t t o a v e r t e x i n X i a n d t h e c l a i m f o l lo w s f r o m o u r i n d u c t i v eh y p o t h e s i s .(2 ) v t - 1 i s a d j a c e n t t o yr .

F o r b y h y p o t h e s i s , I g t - x ( v t ) l < I N t - x ( v t - x ) l _< 2 and so N t - l ( v t ) # N t ( v t ) . T h ec l a i m f o l l o w s .S i n c e ] N t ( v t ) l > 3 w e m a y c h o o s e t h r e e n e i g h b o u r s d l , d : , d a o f v t w i t h d l , d 2 ,d 3 E X t a n d w i t h a ( d ~ ) , n ( d 2 ) , to(d3) d i s t i n c t ; a n d b y ( 2 ) w e m a y t a k e d 3 = v t - 1 . B y( 1 ) t h e r e i s a v e r t e x Vs w h e r e 1 < s < t a n d t h r e e p a t h s P x , P 2, P a o f G , m u t u a l l yd i s j o i n t e x c e p t f o r V s, s u c h t h a t P p h a s e n d s Vs , d p a n d V ( P p ) C_ X t (1 < p < 3).( P o s s i b l y Vs = dl , d2 o r d 3 . ) S i n c e G i s s e r i e s - p a r a l l e l t h e r e i s a p a r t i t i o n ( Y 1 , Y ~, Y 3)o f X t - { V s } s u c h t h a t V ( P p ) C_ Y p U { v s } ( 1 _< p < 3 ) a n d n o e d g e o f G h a s e n d s i nt w o d i f f e r e n t Yp ' S .(3) F o r 2


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392 P . D . S E Y M O U R : C O L O U R I N G S E R I E S - P A R A L L E L G R A P H Sf r o m ( 1 ) t h a t s _< j . S i n c e j < t - 1 i t f o l l o w s t h a t Vt-- 1 ~ 0 8 a n d s o v t - 1 E Y a; a n dh e n c e w e m a y c h o o s e k w i t h j < k < t - 1 m i n i m u m s u c h t h a t v k E Y3. N o w a ( d l ) ,a ( d 2 ) 9 N k ( v ~ ) a n d s o I N k ( v t ) l > 2 , a n d h e n c e I N k ( v k ) l > 2 b y h y p o t h e s i s . B u tk r j s i n c e v j ~ Y a , a n d s o Vk_ 1 ~ Y3 f r o m t h e m i n i m a l i t y o f k , a n d n o t a ll v l , . . . , v kb e l o n g t o Y~ U { V s } s i n c e n e i t h e r v i n o r v j b e l o n g s t o Ya; a n d t h i s c o n t r a d i c t s ( 3 ). |

R e f e r e n c e s[1 ] G . D IR A C : A p r o p e r t y o f 4 - c h r o m a t i c g r a p h s a n d r e m a r k s o n c ri ti c a l g r a p h s , J. LondonMath. Soc. , 2 7 ( 1 9 5 2 ) , 8 5 - 9 2 .[2 ] R . J . D U F F IN : T o p o l o g y o f s e r i e s - p a r a l l e l n e t w o r k s , J. Math. Anal . Appl . , 1 0 ( 1 9 6 5 ) ,

3 0 3 - 3 1 8 .[3] J . E D M O N D S : M a x i m u m m a t c h i n g a n d a p o l y h e d r o n w i th 0 , 1 -v e rt ic e s , J. Res . Nat .Bur. Standards S e c t . B , 6 9 ( 1 9 6 5 ), 8 5 - 9 2 .[4 ] M . G R S T S C H E L , L . L O V ~ ,S Z , a n d A . S C H R IJ V E R : T h e e l li p s o i d m e t h o d a n d i t s c o n s e -q u e n c e s i n c o m b i n a t o r i a l o p t i m i z a t i o n , Combinatorica, 1 ( 1 9 8 1 ) , 1 6 9 - 1 9 7 .[5 ] A . J . H O F F M A N : S o m e r e c e n t a p p l i c a t i o n s o f t h e t h e o r y o f l i n e a r in e q u a l i t i e s t o e x -t r e m a l c o m b i n a t o r ia l a n a ly s is , Com binator ial Analys is , Proc. Syrup. Ap pl . M ath. ,1 0 (1960), 113-127.I. HOLYER: The NP-completeness of edge-colouring, SIAM J. Comput., 10 (1981),7 1 8 - 7 2 0 .

O. MARCOTTE: O n t h e c h r o m a t i c i n d e x o f m u l t i g r a p h s a n d a c o n j e c tu r e o f S e y m o u r( I ) , J. Combinator ial Theory, S e t . B , 4 1 ( 1 9 8 6 ) , 3 0 6 - 3 3 1 .0 . M A R C O TT E: O n t h e c h r o m a t i c i n d e x o f m u l t i g r a p h s a n d a c o n j e c tu r e o f S e y m o u r

( I I ) , p r e p r i n t , ( 1 9 8 7 ).P . D . S EY M O U R " M a t r o id s a n d m u l t i c o m m o d i t y f lo w s , European J. Combinatorics, 2( 1 9 8 1 ) , 257-290.

P. D. S E YM O U R : O n multicolourings f cubic graphs, nd conjectures f Fulkerson nd2 h i t t e , Proc . London Math . Soc ., ( 3 ) 3 8 ( 1 9 7 9 ) , 4 2 3 - 4 6 0 .

P . D . S e y m o u rBellcore, ~ 5 South St., M orristown,New Jer-sey, 07960, U.S.A.

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