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JOURNAL OF COMBINATORIALTHEORY9, 31 5--323 (1971)

The Cross ing Number o f K s , .DANIEL J . K L E I T M A N

The Univers ity o f Calgary, Alberta , Canada, and Massach usettsInsti tute of Technology, Cambridge, MassachusettsCommunicated by the Editor-in-chief

Re c e iv e d J a n u a r y 7 , 19 69

A B S T R A C TS e v e r a l a r g tm a e n t s a r e p r e s e n t e d wh ic h p r o v id e r e s t r i c t i o n s o n t h e p o s s ib l e

n u m b e r o f cr o s s in g s i n d r a win g s o f b ip a r t i t e g r a p h s . I n p a r t i c u l a r i t i s s h o wn th a tc~(Ks.,,) = 4[~n ll 89 - 1)] and er(K~.~) 6[ 89189 -- 1)1.

1 . I N T R O D U C T I O NL e t K i n, . d e n o t e t h e c o m p l e t e b i p a r t i te g r a p h o n s e ts o f m a n d n v e r ti ce s ,

t h a t i s , t h e g r a p h w h o s e e d g e s j o i n e x a c t ly th o s e p a i r s o f v e r ti c es w h i c hb e l o n g o n e t o e a c h s e t. L e t D , ~ , , d e n o t e a d r a w i n g o f K , ~ . , , t h a t is , am a p p i n g o f K i n ,, i n t o t h e p l a n e w h i c h t a k e s t h e v e r t ic e s i n t o d i s t in c tp oi nt s a~, b~ (1 ~ i ~ n, 1 ~ j ~ m ), ca l le d nodes , a n d t h e e d g e s i n t oJ o r d a n a r c s o f f in i te l e n g t h te r m i n a t i n g a t t h e a p p r o p r i a t e n o d e s . Ad r a w i n g i s s a i d t o b e g o o d i f n o t w o a r c s h a v e m o r e t h a n o n e p o i n t inc o m m o n , a n d s u c h a c o m m o n p o i n t i s e i t h er a n o d e w h i c h is a te r m i n a lp o i n t f o r e a c h o f th e a r c s o r i s a crossing. T w o a r cs a i b j , a k b t ( i ~ k , j i f: l )a r e s a i d t o h a v e a crossing i faib~ c a n b e c lo s e d b y a c u r v e c o n n e c t i n g a i t o b jw h i c h i s d i s j o i n t f r o m akb~ a n d s u c h t h a t t h e r e a r e p o i n t s o f akb~ b o t hi n s i d e a n d o u t s i d e t h e c l o s e d c u r v e .

I n a d r a w i n g D , w e d e n o t e b y crD(a~, ak) t h e n u m b e r o f c r os si n gs o fa r c s , o n e t e r m i n a t i n g a t a ~ , t h e o t h e r a t a ~ , a n d b y crD(ai) t h e n u m b e r o fc r o s s i n g s o n a r c s w h i c h t e r m i n a t e a t a ~ :

crD(a i) = ~ crD (a i , a~) .k= l3159 1 9 7 1 b y Ac a d e m ic P r e s s , I n c .

(1 )

582/9/4-~

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316 KLEITMANW e d e f in e t h e cross ing num ber , cr (D) , o f D t o b e t h e n u m b e r o f cr o s si n g si n D : cr(D) = ~ ~ , c rD(ai ,ak) . (2 )

i ~ l k = i + lT h e cross ing n um be r , cr (K~.,~) , i s d e f i n ed t o b e t h e m i n i m u m c r o s s in gn u m b e r a m o n g g o o d d r aw i ng s o f K . . . . . F o r g o o d d r aw i n gs , c r D ( a i , a i ) = 0a n d

cr (D) = 89~ crD(ai). (3 )A l t h o u g h o n e c a n h a v e d r a w i n g s i n w h i c h a r c s a r e p e r m i t t e d t o i n t e r s e c tt w o o r m o r e t i m e s , i t is e as y to v e r i f y t h a t t h e c r o s s i n g n u m b e r o f s u c h ad r a w i n g w i ll e x c e e d t h a t o f a s u i t ab l e g o o d d r a w i n g .

I n t h e r e m a i n i n g s e c ti o n s a n u m b e r o f a r g u m e n t s a r e p r e s en t e d , e a c h o fw h i c h g i v e s s o m e r e l e v a n t i n f o r m a t i o n c o n c e r n i n g cr(K~.n) . I n c o m b i -n a t i o n t h e se s h o w t h a t f o r m ~ 6 ,

cr(K,~,,~) = [89189 - - 1 ) ] [ 891 89 1)1 . (4)W e s h a l l r e f e r t o ( 4 ) as Z a r a n k i e w i c z ' s c o n j e c t u r e a n d d e n o t e t h e r i g h tm e m b e r b y Z ( m , n ).

T h e h i s t o ri c a l b a c k g r o u n d o f th i s p r o b l e m i s g i v e n i n [ 1].

2 . C O N S T R U C T I O N

Z a r a n k i e w i c z [2 ] g a v e a d r a w i n g o f K i n,, w h i c h d e m o n s t r a t e s t h a tcr(Km ,~)

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THE CROSSING NUMBER OF K s , n 317

H e n c e w e c a n o b t a i n b y a r i t h m e t i c o p e r a ti o n s t h a tcr (Km,~8_l ) - : - Z (m , 2s - - 1) imp l i e s c r (K m . 2 ~ = Z ( m , 2s) , (7)

f o r a l l n , a n d t h a tcr (Ks .~ . ) = Z (5 , 2s) im plie s cr(5,2~+LYl) >~ Z (5 , 2s + 1) - - 2 , (8)

f o r s ) 2 , a n d m o r e g e n e ra l l y t h a tcr (Kz~+a.2 ,) = Z (2 r + 1, 2s)

r 1m p l i e s cr(K~r+l.Zs+l) ~ Z (2 r + 1, 2 s + 1 ) - - L - ~ - - ~ T J . ( 9)A n o t h e r i m p l i c a t i o n o f t h i s k i n d , o b t a i n e d b y s u p p r e s s i n g a~ i n D .. .. . i s

c r D (a i ) + c r ( K . . . _ x ) ~ c r ( D m , . ) . (10)

4. ZARANKIEWICZ'S ARGUMENTI f, f o r s o m e a i a n d a ~ , cr~ (a ~ , a~ ) = O , t h e n w e c a n c o n c l u d e t h a t

c r( D m ,~ ) ~ c r ( K . . . 3 ) + ( n - - 2) c r ( K ~ , a ) ,s in c e t h e c r o ss i n g s w h i c h i n v o l v e a i , a~ a n d e a c h o t h e r a m u s t b e a t l e a stcr(Km,3) i n n u m b e r , a n d t h o s e n o t in v o l v i n g a~ , ak n u m b e r a t le a stc r ( K . . . 2)- A r i t h m e t i c m a n i p u l a t i o n s a g a i n s h o w t h a t c r ( K m , n _ ~ ) =Z ( m , n - 2 ) a n d c r ( K m . 3 ) - ~ Z ( m , 3 ) t o g e t h e r i m p l y t h a t u n d e r t h e s ec i r c u m s t a n c e s , c r ( D , n , . ) > ~ Z ( m , n ) .

T h i s a r g u m e n t a l o n e y i e l d scr (Ka,n) ~ - Z (3 , n) = [ 89 - - 1)], (11)

s inc e i f crD (ai , ak ) -7 0 f o r a n y i , k , t h e n cr(D3.,~) ~> (~) > Z (3 , n) .I n t h e s e q u e l w e w i ll a s s u m e Z a r a n k i e w i c z ' s c o n j e c t u r e as a n i n d u c ti v e

h y p o t h e s i s a n d a s a r e s u lt o f t h is a r g u m e n t w e c a n n o w l i m i t o u r c o n s i d -e r a t i o n t o d r a w i n g s i n w h i c h crv (a i , ak ) ~ /~ 0 f o r a n y i a nd k , i :7~ k .

5. PARITY ARGUMENT

S i n c e w e a r e d e a l i n g w i t h d r a w i n g s i n t h e p l a n e , w e c a n c o n t i n u o u s l yd e f o r m a n y d r a w i n g i n t o a n y o t h e r . G i v e n t w o d r a w i n g s D , D ' o f K m ,n w ef ir st i d e n t i fy t h e n o d e s o f D w i t h t h o s e o f D ' a n d t h e n d e f o r m t h e a r c s o f D

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318 KLEITMANo n e b y o n e i n t o t h o s e o f D '. T h e p a r i t y o f th e n u m b e r o f cr o ss in g s o f a na rc aibj i n D i s o n l y a l t e r e d w h e n a c r o s s in g a r c akb~ i s d e f o r m e d p a s t at e r m i n a l n o d e a i o r b ~ ., o r w h e n asb~ i s d e f o r m e d a c r o s s ( p a r t o f ) a na r c a s b j . S u p p o s e t h e a r c asb~ is d e f o r m e d p a s t b j . T h e n t h e p a r it y o f th en u m b e r o f c ro s si ng s o f asb~ w i t h e a c h a r c t e r m i n a t i n g a t b j w i ll b e c h a n g e d .H o w e v e r , i f b e f o r e d e f o r m a t i o n t h e d r a w i n g w a s g o o d , i n e i th e r ca s e t h ed e f o r m a t i o n i n tr o d u c e s a n i n t er s e c ti o n o f asb~ w i t h a s b j . T h i s i n t e r s e c t i o nw i l l n o t b e a c r o s s i n g i f w e i n t e r c h a n g e t h e s e g m e n t s j o i n i n g i t t o a s 9( S e e F i g u r e 1 , i n w h i c h t h e t w o p e c k e d a r c s asb t a r e d e f o r m a t i o n s o f e ac h

I

/ /"-3/ \

c i b~F I G U R E 1

o t h e r ; t h e o n e w h i c h c r o s s e s a kb ~ , a n d a k b j i t s e l f , a r e r e p l a c e d by thed o t t e d a r c s akbt a n d asbs.

T h e r e f o r e , i n d e f o r m i n g akb~ p a s t b~ w e h a v e c h a n g e d t h e p a r i t y o f t h en u m b e r o f c r o ss i n g s o n e v e r y a r c t e r m i n a t i n g i n b j , e x c e p t a s b j . T h u s , i f nis o d d , t h e p a r i t y o f t h e t o t a l n u m b e r o f c r o s si n g s is u n c h a n g e d .

W e c o n c l u d e t h a t t h e p a r i t y o f t h e c ro s s in g n u m b e r o f tw og o o d d r a w i n g s o f K2r+ l , 2 s+1 i s t h e s a m e . F r o m t h e c o n s t r u c t i o n o fS e c t i o n 2 , i t f o l l o w s t h a t cr(K2~+l,4u+a) s e ve n , wh i l e cr(K4,+3.4u+3) s o d d .I t a l s o f o l l o w s t h a t , f o r a g o o d d r a w i n g D o f K 4t+ l,2 s+ l ( o r o f K4t+3.~s+a),c rD ( a i) + e r D ( a k ) - c r D (a i, a k ) i s e ve n ( o r odd , r e spe c t ive ly ) , s i nc e th i sn u m b e r r e p r e s e n t s t h e d i ff e re n c e b e t w e e n t h e c r o s s in g n u m b e r o f D a n dt h a t o f th e d r a w i n g o b t a i n e d b y s u p p r e s s in g n o d e s a~ a n d a s i n D .

6 . S P E C I A L C O U N T I N G A R G U M E N T

L e t n b e t h e l e a s t i n t e g e r, i f a n y , s u c h t h a t cr (Ks . . ) ~ Z( 5 , n ) . I mp l i c a -t i o n s ( 7 ) a n d ( 8 ) o f S e c t i o n 3 s h o w t h a t n i s o d d a n d

c r ( K s , ~ ) > / Z ( 5 , n) - - 2 . (12)T h e p a r i t y a r g u m e n t o f S e c t i o n 5, t o g e t h e r w i t h t h e c o n s t r u c t i o n o fS e c t i o n 2 , t h e n i m p l y e q u a l i t y i n ( 1 2 ) .

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THE CROSSING NUMBER OF g 5 , n 319

A n y d r a w i n g D o f K s,~ , s a t i s fy i n gc r ( D ) = Z ( 5 , n ) - - 2

m u s t a l s o s a t i s f y( 1 3 )

~ , c r D ( a i ) = 2 Z ( 5 , n ) - - 4 --~ n ( 2 n - - 4 ) - - 2 .i=1

A l s o , b y ( 1 0 ) , f o r e a c h i,c r D ( a i ) . ~ z ( 5 , n ) - 2 - Z ( 5 , n - 1 ) = 2n - 4 .

T h e r e f o r e , f o r a ll b u t t w o v a l u e s o f i, w e m u s t h a v e c rD(a O = 2 n - - 4 . I fcrD(a~) h a s t h e s a m e p a r i t y f o r e v e r y i, t h e n t h e c o n c l u s i o n o f S e c t i o n 5s h o w s t h a t c r D ( a i , a k ) i s e v e n f o r a l l i a n d k . B y t h e a r g u m e n t o f S e c t io n 4 ,c r D ( a i , a k ) > ~ 1 f o r i =/= k , s o i n t h i s c a s e c r D ( a i , a ~ ) >~ 2 ( i ~ k ) a n d ( 2 )g i v e s

c r ( D ) ~ 2 ( 2 ) > Z ( 5 , n ) ,w h i c h c o n t r a d i c t s ( 13 ) , o u r in i t i a l a s s u m p t i o n c o n c e r n i n g D .

T h e o n l y o t h e r p o s s i b i l i t y is t h a t f o r e x a c t ly t w o v a l u e s o f i , w h i c h w et a k e t o b e 1 a n d 2 , crD(a i) = 2 n - - 5 ( i = 1 , 2 ) , w h i l e e r D ( ai ) = 2 n - - 4(3 ~< i ~< n ) . T h e c o n c l u s i o n o f S e c t i o n 5 h e r e i m p l i e s t h a t erD(a~, ak ) >~ 2f o r k > i >~ 3 . T h e o n l y w a y i n w h i c h t h e s e c o n d i t i o n s c a n b e s a t i s fi e d i sa s f o l lo w s :

c r ~ ( a i , a ~ ) = 2 , 3 < ~ i < k < ~ n , ( 1 4 )c rD(a ~ , a~) = 1 , i = 1 , 2 ; 3 ~ k ~< n , (15 )c r D ( a l , a ~ ) = n - - 3 . ( 1 6 )

T h e a r g u m e n t s o f t h e n e x t th r e e s e c t io n s w i ll s h o w t h a t t h e se c o n d i t i o n sc a n n o t , i n f a c t , h o l d .

7 . CONSTRUCTIVE ARGUMENTG i v e n a d r a w i n g D o f K , , ~ w e c a n c o n s t ru c t a d r a w i n g o f K i n , n + 1 i n t h e

f o l lo w i n g m a n n e r . C h o o s e a n y n o d e , a ~ , i n D , a n d i n s e r t a n e w n o d e , a ~ + 1 ,n e a r a i ( e .g . , i n a c i rc l e , c e n t e r a i , a n d r a d i u s s u f f i c ie n t l y s m a l l t h a t i tc o n t a in s n o p o i n t s o f a r c s o t h e r t h a n t h o s e t e r m i n a t i n g a t a i ) . J o i n a , + l t oe a c h o f t h e b j b y a n a r c w h i c h f o l l o w s e s se n t ia l ly t h e s a m e p a t h a s a ~ b j . W e

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320 KLEITMANc a n d o t h is s o t h a t t h e c r o s si n g s i n t r o d u c e d b e t w e e n a r c s an+xb~ 1 ~ j ~ m )a n d akb~(i =~ k % n) w i ll n u m b e r crD(ai a n d t h o se b e t w e e n a,+lbj) a n d aib~c a n b e m a d e t o n u m b e r Z ( m , 3 ) ( c f . f o r m u l a ( 1 8 ) b e l o w ) . T h e c o n s t r u c t i o no f S e c t io n 2 c a n b e c o n s i d e r e d a s a s e q u e n c e o f c o n s t r u c t i o n s o f th i s t y p e .

W e m a y i m m e d i a t e l y c o n c l u d e t h a t t h e c o n d i t i o n s ( 1 4) t o ( 16 ) c a n n o t b es a ti s fi e d f o r n ~ 7 , f o r , u p o n s u p p r e s s i n g n o d e a x , o n e w o u l d o b t a i n ad r a w i n g D ' o f K 5 ,, -1 s a ti s fy i n g crD, ( a i , ak ) ---- 2 (3 ~ i < k ~< n) ,crD,(a2, a k ) = 1 (3 ~ k ~ n) . T h e a b o v e c o n s t r u c t i o n a l lo w s u s t or e i n s e r t n o d e a l n e a r a 2 t o p r o d u c e a d r a w i n g D " of K s ,~ s a t i s f y i ng ( 14 ), ( 15 )a n d c r D . ( a l , a s ) = Z ( 5 , 3 ) ~ - 4 . S u c h a d r a w i n g w o u l d v i o l a t e c o n d i t i o n(12) .

T h e p r o b l e m o f co n f i r m i n g Z a r a n k i e w i c z ' s c o n je c t u re f o r K s , . is th u sr e d u c e d t o t h e c a s e s n = 5 a n d n = 7 .

8. PERMUTATION ARGUM ENT

L e t D b e a d r a w i n g o f K , , , , a n d l e t a i b e a n o d e . W e m a y a s s o c ia t e w i t ha , a s e t ~r~ o f p e r m u t a t i o n s o f t h e i n t e g e r s 1 ~ j ~ m b y t h e f o l l o w i n gp r o c e d u r e . D r a w a c l o s ed c u r v e r o u n d a , , s u ff ic ie n tl y s m a l l t h a t i t i n t e r-s e c t s e a c h a r c aib~ e x a c tl y o n c e in a p o i n t P j . T h e m e m b e r s o f rri a r e th ec l o c k w i s e o r d e r s o f t h e l a b e l s j o f t h e s e p o i n t s . T h e m m e m b e r s o f rr~ a r ec y c l ic t r a n s p o s i t i o n s o f e a c h o t h e r . W e d e n o t e b y 7 7, t h e s e t o f a n t i -c l o c k w i s e o r d e r s o f t h e l a b e l s .

I f crD(ai , a~) = 0, th en 7ri = 77~.I f t h e t r i a d JlJ2Ja o c c u r s i n t h e s a m e o r d e r i n 7 q a n d r r k , i t i s n e c e s s a r y

t h a t t h e r e b e a t l e as t o n e c r o s s in g a m o n g t h e s i x a r c s a~b~, akb j .M o r e g e n er al ly , l et X~,k b e t h e m i n i m u m n u m b e r o f in t e rc h a n g e s a m o n g

a d j a c e n t e l e m e n t s o f a m e m b e r o f ~-i t o o b t a i n a m e m b e r o f ~'k t h e n i tf o l l o w s i n d u c t i v e ly t h a t

cro(ai , ak) ~ X~,k. (17)I n t he c a s e r ri = 7 r k, ( 17 ) i m p l i e s t h a t

cro(ai , a~) ~ Z (m , 3). ( 1 8 )I n t h e c a s e K s , , t h e r e a r e 4 ! p o s s i b l e s e t s z q . I n o r d e r t o r e a l i z e t h e

r e q u i r e m e n t s ( 1 4) to ( 1 6) f o r n = 5 , t h e p e r m u t a t i o n s r ri w o u l d h a v e t osa t i s fy the co nd i t io ns Xl ,k = X~,k = 1 (k -- -- 3 , 4 , 5 ) an d Xi ,~ ~ 2( o t h er w i s e ). W e m a y w i t h o u t l o s s o f g e n e r a l i ty r e p r e s e n t z?1 as (12345) ;r r3, r r4 , a n d zr5 w o u l d t h e n e a c h h a v e t o d i f fe r b y e x a c t l y o n e a d j a c e n t

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THE CROSSING NUMBER OF Ks , n 321

i n t e r c h a n g e f r o m r71 , a n d i n v i r t u e o f ( 18 ) t h e y m u s t b e d i s ti n c t . T h e y a r et h u s t o b e c h o s e n f r o m

(21345) , (13245 ) , (12435) , (12354 ) , an d (52341) . (19)T h e r e i s n o p e r m u t a t i o n s e t 7 7~ , d i s t in c t f r o m 771 , w h i c h d i f fe r s b y o n l y o n ea d j a c e n t i n t e r c h a n g e f r o m t h r e e o f (1 9). I n c o n s e q u e n c e , t h e c o n d i t i o n s ( 14 )t o (1 6 ) c a n n o t b e m e t f o r n = 5 .

F o r n = 7 , c o nd i t i on s ( 14 ) t o ( 16 ) i m p l y t ha t X 1 ,2 = 4 , X 2 ,k = 1( 3 ~< k ~ 7 ) a n d X i ,k ~< 2 ( o t he r w i s e ) . I f w e r e p r e s e n t 7?2 b y ( 1 23 4 5 ) t h e n~r~ (3 ~< k ~ 7 ) m u s t a g a i n b e d i s ti n c t a n d c h o s e n f r o m t h e p e r m u t a t i o n s( 1 9 ) . H e n c e t h e y m u s t b e ( 1 9 ) .

9 . H o c u s - P o c u s ARGUMENTW e n o w s h o w t h a t i n a g o o d d r a w i n g D o f K s, 6 w i t h n o d e s a k

(2 ~ < k ~ < 7 ) , b j (1 ~ < j ~ < 5 ) , a n d w i t h 7?2 = ( 12 34 5) , rr 3 = ( 2 1 3 4 5 ) ,~r4 = (13245) , r r5 = (124 35) , ~ 'n = (123 54) , r r 7 = (523 41) , i t i s n o tp o s s i b l e t o s a t i s f y c rD ( a 2 , a k ) = 1 ( 3 ~ k ~< 7 ) a n d c r o( a i, a k ) = 2( 3 ~ < i ~< 7 ). I f w e c o n s i d e r t h e p o r t i o n o f D , i n v o l v i n g o n l y a r c s t e r m i -n a t i n g a t a s a n d a 3 , u n d e r t h e c i r c u m s t a n c e s g i v e n h e r e, i t m a y b e d r a w ni n t w o p o s s i b l e w a y s , w i t h a3b2 c r o s s i n g a2b1 , o r w i t h a3bl c r o s s i n g a2b2.W i t h o u t l o ss o f g e n e r a l it y , w e a s s u m e t h e f o r m e r ( F i g u r e 2 ).

G2

+

(13

b5

FIGURE 2

I n o r d e r t h a t c r o ( a 2 , a~) = i and crD(a3, ak) = 2 (4 ~< k ~< 7) , ea cho f th e n o d e s a k ( 4 ~< k ~< 7) m u s t l ie i n o n e o f th e r e g i o n s m a r k e d I a n d I Ii n F i g u r e 2 . F o r e x a m p l e , i f a 4 w e r e t o l i e i n r e g i o n I I I , t h e a r c s a4b~( j = 1, 2 , 3 ) w o u l d e a c h h a v e t o c r o s s a n a r c o f t h e c i r c u i t a2b4a3bsa~,w hi l e a4bl a n d a4b2 a l s o c r o s s t he c i r c u i t aJFa3b3a2. M o r e o v e r , n o a r c

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8/9

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9/9

THE CROSSlNG NUMBER OV Ks,n 323and the counti ng argu ment of Section 3 now enables us to establish thelower boun d

c r ( K , , n ) ~ ~ m ( m - - 1)[89 - - 1)], m ~ 5.For example, this, together with the parity argument, gives cr(Kv,v) ~ 77,so that cr(KT,~) is 77, 79, or 81. Which?

ACKNOWLEDGMENTI am indebted to R. K. Guy for writing this paper.This research was supported in part by NSF contract GP 90003.

REFERENCES1. R. K. GuY, The Decline and Fall of Zarankiewicz's Theorem, Methods of Proofin Graph Theory (Proc. Ann Arbor Conference 1968), Academic Press, New York,1969.2. K. ZARANKIEWICZ,On a Problem of P. Tur~in concerning Graphs, Fund. Math. 41

(1954), 137-145.