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Comput Math Applic Vo l. 15, No. 4, pp. 277-289, 1988 009%4943/88 $3.00+0 .00

Printed in Great Britain. All rights reserved Copy right © 1988 Pergam on Press pie

A S U R V E Y O F T H E T H E O R Y O F H Y P E R C U B E G R A P H S

F R A N K H A R A R Y

Co m p u t in g R e se a rc h La b o ra to ry , N e w M e x ic o S ta t e U n iv e r s it y , La s Cru c e s, N M 8 8 0 0 3 , U .S .A .

J O H N P . H A Y ES a n d H O R N G - J Y H W u

A d v a n c e d Co m p u te r A rc h i t e c tu re La b o ra to ry , U n iv e r s i t y o f Mic h ig a n , A n n A rb o r , M I 4 81 0 9, U .S .A .

A lm t ra c t - -We p re se n t a c o m p re h e n s iv e su rv e y o f t h e t h e o ry o f h y p e rc u b e g ra p h s . Ba s ic p ro p e r t i e s r e l a te d

to

d i s t an c e , c o lo r in g , d o m in a t io n a n d g e n u s a re r e v ie w e d. Th e p ro p e r t i e s o f t h e n -c u b e d e f in e d b y i t s

subgrap hs a re considered next , inc luding th ickness, coarseness, Ham il tonian cyc les and induced pa ths an d

cyc les . F ina l ly , var ious embedding and packing problems a re d iscussed , inc luding the de te rmina t ion of

th e c u b i c a l d im e n s io n o f a g iv e n c u b i c a l g ra p h .

I . I N T R O D U C T I O N

T h e

n-cube

o r

n-dimensional hypercube Q.

i s def ined recurs ive ly in t e rms of t he car t es i an product

[1 , p . 22] of two graphs as fo l lows:

Q =Ks

Q = K 2 x Q _ .

(1)

T h u s t h e n -cu b e , o r m o re b r i e f l y t h e

cube Q.

m ay a l so b e d e f i n ed a s a g r ap h w h o se n o d e s e t V ,

cons i s t s o f the 2 n-d im ens ion a l bo olea n vec tors , i. e., vec tors wi th b inary coo rd ina t es 0 or 1 , wh ere

t w o n o d es a r e a d j acen t w h en ev e r t h ey d i f fe r i n ex ac tl y o n e co o rd i n a t e . F i g u re 1 sh o w s t h e n -cu b e s

fo r n ~< 3 w i th ap p ro p r i a t e b o o l ean v ec t o r s a s n o d e l ab e l s. C u b e g rap h s h av e b een m u ch s t u d i ed

i n g rap h t h eo ry . I n t e r e s t i n h y p e rcu b es h a s b een i n c rea sed b y t h e r ecen t ad v en t o f m ass i v e ly p a ra l le l

co m p u t e r s w h o se s t ru c t u re i s t h a t o f t h e h y p e rcu b e [2 , 3 ]. T h is n o t o n l y p ro v i d e s p o t en t i a l

ap p l i c a t io n s fo r t h e ex i s t in g t h eo ry , b u t a l so su g g es t s so m e n ew a sp ec t s o f cu b es t h a t d e se rv e s t u d y .

W e su rv ey t h e g rap h - t h eo re t ic l i t e r a tu re o n n -cu b es , an d p re sen t a su m m a ry o f t h e m a j o r k n o w n

resu l t s . S o m e n ew p ro b l em s , m o t i v a t ed i n p a r t b y p a ra l l e l co m p u t i n g co n s i d e ra t i o n s , a r e a l so

p re sen t ed . E m p h as i s is p l aced o n t h e sp eci a l p ro p e r t i e s an d su b g ra p h s o f cu b es , a s w e ll a s t h e

p r o b l e m s o f e m b e d d i n g a n d p a c k i n g g r a p h s i n c u b e s . M o s t o f t h e n o t a ti o n u s e d m a y b e f o u n d

i n H a ra ry [1 ]. A g rap h G = (V , E ) h a s p =

I v I

n o d es an d q =

I I

edges , and i s sa id to have

order

p a n d

size q.

Thus , t he o rde r o f Q , i s 2 and i ts s ize i s n 2 - ' .

2 . B A S I C P R O P E R T I E S

W e b eg i n b y su rv ey i n g so m e i n v a r i an ts o f h y p e rcu b es r e la t ed t o t h e d i s tan ce b e t w e en t w o n o d es .

T h i s d i s tan ce i s t h e n u m b er o f co o rd i n a t e s i n w h i ch t h e co r r e sp o n d i n g b o o l ean v ec t o r s d i ff e r. T h e

I 1 11

O 1

1 1 1 1 1

1

1 2 Q3

Fig . . n -cube grap hs f or n = 1 , 2 , 3 .

2 7 7


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278 FRANK HARARY

e t a l

d i a m e t e r d ( G )

o f a g r a p h G is th e m a x i m u m d i s ta n c e b e t w e e n a n y p a i r o f n o d e s; o b v i o u s l y

d ( Q , ) = n .

T h e

total distance

o f g r a p h G w i t h n o d e s e t V = { v ~, v2 . . . . vp} i s

t d ( G ) = Z

d ( v , , O

I~


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A survey of the theory of hypercube graphs 279

n u m b e r o f G . T h e node--edge, edge-node a n d edge -edge domina t ion numbers c a n b e s i m i l a r l y

d e f i n e d a n d a r e d e n o t e d b y ~ 0~ , cq0 a n d ~ t~ , r e s p e c t i v e l y . F o r e x a m p l e , ~ 1 0 ( G ) i s t h e m i n i m u m

n u m b e r o f e d g e s t h a t d o m i n a t e a ll n o d e s o f G . T w o n o d e s o r t w o e d g e s a r e

independent

i f t h e y

a r e n o t d o m i n a t e d b y e a c h o t h e r . A s et S o f n o d e s o r e d g e s o f G is i n d e p e n d e n t i f a n y p a i r o f n o d e s

o r e d g e s in S a r e i n d e p e n d e n t i n G . T h e m a x i m u m c a r d i n a l i t y o f s u c h a se t is t h e

node independence

number f lo

o r

edge independence number i l l .

G a l l a i [ 4 ] p r o v e d t h a t f o r a n y n o n t r i v i a l g r a p h G ,

c% + fl0 = P = ~lo + ill .

T h i s r e s u l t l e a d s d i r e c t l y t o g e n e r a l f o r m u l a s f o r ~ 0 a n d ct0~ i n a c u b e , a n d h e n c e f o r f l0 a n d f it ,

c% (Q , ) = ~ o(Q ,) = f lo (Q,) = i l l (Q , ) = 2 -~. (4)

T h e f o l l o w i n g r e l a t e d r e s u lt c o m m u n i c a t e d t o u s b y Q . F . S t o u t i s k n o w n f o r C to o(Q ,). F o r t h e

spe c i a l c a se n = 2 k - 1, we ha ve c t00(Q,) = 2 /n + I . T h e m a x i m u m n u m b e r o f n o d e - d is j o in t c o p ie s

o f a s ta r S = K ( 1 , 2~ - l ) t h a t c a n b e e m b e d d e d i n t o

Q ,

is 2 - k . N o t e t h a t t h e c e n t e r o f S d o m i n a t e s

a ll o t h e r n o d e s o f S . I f th e m a x i m u m n u m b e r o f n o d e - d i s jo i n t c op i es o f S i s e m b e d d e d in a

h y p e r c u b e , t h e n t h e i r c e n t e r s f o r m a m i n i m u m d o m i n a t i n g s e t . H o w e v e r , = 0 0 ( Q , ) i s n o t k n o w n

when n =~

2 k -

l .

T h e e x a c t d e t e r m i n a t i o n o f C q l( Q ,) is e v e n m o r e d i f fi cu l t. O b v i o u s l y ~ H ( Q : ) = 2 . T h e h e a v y l in e s

i n F i g. 2 m a r k m i n i m a l s e t s o f e d g e s t h a t d o m i n a t e a ll ed g e s o f Q 3 a n d Q 4, p r o v i n g t h a t 0 ~ l l Q 3 ) = 3

a n d 0 q j( Q 4 ) = 6 . S t o u t h a s a l s o i n f o r m e d u s o f t h e f o l l o w i n g b o u n d s f o r n ~> 3 :

n2 /(3n 1) ~< ~l l (Q , ) ~< 3( 2 -3 ) • (5)

E q u a t i o n ( 4 ) st a te s t h a t fl~ ( Q , ) = 2 - 1, s o t h a t t h e m a x i m u m n u m b e r o f i n d e p e n d e n t e d g e s i n Q ,

is o n e - h a l f t h e n u m b e r o f n o d e s. I n [5 ], t h e m i n i m u m n u m b e r o f ed g e s in a m a x i m a l i n d e p e n d e n t

s e t , d e n o t e d b y

f l~-(G),

w a s in t r o d u c e d . C l e a rl y , f o r a n y g r a p h G , ~ ( G ) =

f l~-(G).

T h u s F i g . 2 ( a )

a l s o i l lu s t r a t e s t h e f a c t t h a t f l i - ( Q 3 ) = 3.

W e c a n e x t e n d t h e d e f i n i ti o n o f d o m i n a t i o n t o a r b i t r a r y s u b s e ts V~ a n d Vz o f

V ( G ) .

W e s a y t h a t

V I d o m i n a t e s V2 i f f o r a n y y e l iE , t h e r e e x i s ts x E V~ w h i c h d o m i n a t e s y . A D-par t i t ion o f G i s

t h e n d e f i n e d a s a p a r t i t i o n o f

V ( G )

i n t o d o m i n a t i n g s e t s . A n i n v a r i a n t s i m i l a r t o t h e d o m i n a t i o n

n u m b e r w a s i n t r o d u c e d i n [6 ]. I t i s c o n c e r n e d w i t h t h e o r d e r o f th e D - p a r t i t i o n i n s te a d o f t h e o r d e r

o f t h e d o m i n a t i n g s et. T h e

d o m a t i c n u m b e r r ( G )

is t h e m a x i m u m o r d e r o f th e D - p a r t i t io n s o f G .

F i g u r e 3 d i s p l a y s D - p a r t i t i o n s o f m a x i m u m o r d e r 4 , w h e r e t h e n o d e s a r e l a b e le d a s r e si d u e cl as s es

r o o d 4 . C l e a r l y , D - p a r t i t i o n s c a n b e e q u i v a l e n t l y d e f i n e d b y a c o l o r i n g p r o c e d u r e , c a l l e d

domat ic

coloring,

w h e r e e a c h n o d e c o l o r e d b y a c e r t a in c o l o r i s a d j a c e n t t o n o d e s c o l o r e d b y a l l t h e o t h e r

c o l o r s , a n d T ( G ) is t h e m a x i m u m n u m b e r o f c o l o r s u s e d . I n [6 ], t h e t h e o r y o f d o m i n a t i o n o f a

g r a p h is r e v i e w e d in d e t a i l , a n d i t i s s h o w n t o b e r e l a t e d t o s e v e r a l f ie l d s o f s t u d y . F o r e x a m p l e ,

a m a t c h i n g o f a g r a p h G c o r r e s p o n d s t o a n i n d e p e n d e n t d o m i n a t i n g s e t o f n o d e s in t h e li ne g r a p h

o f G . O n e o b v i o u s o b s e r v a t i o n i s th a t

z ( G ) ~ <

6 ( G ) + 1 . T h e f o l l o w i n g r e s u lt o n r ( Q , ) is d u e t o

Z e l i n k a [ 7 ] : f o r a l l p o s i t i v e i n t e g e r s k ,

r(Qz~_

,) = ~ (Q2k) = 2 ~. (6)

101 111

100 110

(a)

Fig. 2. Minim al sets of edges (heavy

1001 1011

N O 1 0101 0111 1 1 ~

1000 10T ~

b )

fines) that dom inate all edges of Q3 and Q4.


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2 3

l o

3

a)

1 0

2 8 0 F R A N K H A R A R Y e t a l

0 1

b)

Fig. 3. Do m atic 4-colorings of Q3 and Q4.

A l t h o u g h t h e d o m a t i c n u m b e r s f o r cu b e s o f o t h e r d i m e n s i o n s a re n o t k n o w n , Z e l i n k a [7]

co n j ec t u r ed t h a t i f n + 1 i s n o t a p o w er o f 2 , t h a n T ( Q , ) = n .

3. T O P O L O G I C A L I N V A R I A N T S

T h e genus 7 ( G ) is t h e m i n i m u m n u m b e r o f h a n d l e s w h i ch m u s t b e a d d e d t o a s p h e re s o t h a t

G c a n b e e m b e d d e d i n t h e r e s u l t i n g s u rf a c e w i t h n o e d g e s c r o ss i n g . T h u s a g r a p h G is planar w h e n

( G ) = 0 , an d i s c a l l ed

toroidal

i f Y G ) = 1 . T h e g e n u s o f a cu b e w as f i r s t f o u n d b y R i n g e l [8 ] an d

l a t e r w a s i n d e p e n d e n t l y r e d i sc o v e r e d b y B e i n e k e a n d H a r a r y [ 9 ]. I t is e v i d e n t f r o m F i g . 2 t h a t

? (Q 3 ) = 0 a n d ? ( (2 4) = 1 . F r o m E u l e r ' s ch a r ac t e r i s t i c e q u a t i o n f o r s p h e r i c a l p o l y h ed r a , w e k n o w

t h a t f o r a p o l y h e d r o n o f g e n u s ? w i t h V n o d e s , E e d g e s a n d F f a c e s ,

V - E + F = 2 - 27. (7)

I f G is a c o n n e c t e d g r a p h o f g e n u s ? a n d h a s n o t r ia n g l e s th e n , a s m e n t i o n e d i n [9 ], t h e f o l l o w i n g

i n eq u a l i t y i s i m p l i ed b y eq u a t i o n ( 7 ) :

q p - 2

7 ( G ) > t

4

H e n c e ,

n 2 - I 2 - 2

( Q . ) / > T T = ( n - 4 ) 2 - 3 + l '

a n d w e d e n o t e t h e l a t te r e x p r e s s io n b y ~ ,. E m b e d d i n g s o f Q , i n a n o r i e n t a b l e s u r fa c e o f g e n u s 7,

a r e co n s t r u c t ed i n [ 8 , 9 ], p r o v i n g t h a t 7 ( Q , ) ~ < 7 , , t h e r e f o r e

7 (Q ,) = (n - 4)2 -3 + 1. (8)

T h e crossing number v G) o f G i s d e f i n e d a s th e m i n i m u m n u m b e r o f p a ir w i s e i n t e rs e c t io n s o f

i ts e dg e s w h e n G i s d r a w n i n t h e p l a n e . T h e d e t e r m i n a t i o n o f th e e x a c t v a l u e o f

v G)

i s k n o w n

t o b e N P - c o m p l e t e . I t h a s b e e n m u c h s t u d i e d fo r c o m p l e t e g r a p h s a n d c o m p l e t e b i g r a p h s b u t o n l y

u p p e r b o u n d s ( w h i ch m o s t l i k e l y g i v e t h e ex a c t v a l u e s ) a re e s t ab l i s h ed [3 , 1 0 ]. A l l t h a t i s k n o w n

is

v (Q3) = 0

a s Q 3 i s p l an a r , an d

v O , ) = 8

a s i l l u s t r a t ed i n F i g . 2 ( b ). I t i s e a s y t o d r aw Q 5 i n t h e p l an e w i t h 5 6 c r o ss i n g s ; h en ce

v (Qs) ~< 56.

T h e d e t e r m i n a t i o n o f t h e e x a c t v a l u e o f v ( Q , ) f o r g e n e r a l n is a f i e n d i sh l y i n t r a c t a b l e p r o b l e m .


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6/13

282

FR NK

H R RY

e t a l .

Table I. The numbers H, and h, of

hamiltonian cycles in an unlabeled and

in a labeled cube Q,

n H n h.

2 I 1

3 1 6

4 9 1344

5 2 7 5 , 0 6 5 906,545,760

The cited value for h s is from A. Bell

and P. Hallowell, Crawling round

a cube edge, C o m p u t i n g U . K . ) ,

p. 9 (Feb. 1973). The value o f H 5

was determined by D. Russell

in June 1987; his article giving

the method of solution will be

published.

Table 2. The length s, of

snakes and c, of coils in Q,

n s n c n

1 1 2

2 2 4

3 4 6

4 7 8

5 13 14

6 26 26

n > i 6 , n o e x a c t v a lu e s a r e k n o w n b u t D o u g l a s [ 14 ] a c t u a l l y f o u n d t h e l o w e r a n d u p p e r b o u n d s i n

i n e q u a l i t y ( 1 1 ) ; s e e D i x o n a n d G r a h a m [ 1 5 ] .

2 - i - t ( 1 3 4 4 ) 2 -4 . n . 2 { 2 - 2 - ~ - ] < ~ h ( Q . ) < ~ L n ( n - 1 ) / 2 3 2 - ' - 2 t - ' - ' ° ' : ( 1 1)

\ i=

B y d e f i n i t i o n H i s a n i n d u c e d s u b g r a p h o f G i f f o r a n y u , v • V ( H ) , i f u a n d v a r e a d j a c e n t i n

G t h e n t h e y a r e a l s o a d j a c e n t i n H . T h e i n d u c ed p a t h s a n d i nduc e d c y c l e s o f n - c u b e s a r e s u b g r a p h s

o f s ig n i f ic a n c e t o c o d i n g t h e o r y a n d r e l a t e d a r e a s [ 1 6 -1 8 ] . A n n - s n a k e i s a l o n g e s t i n d u c e d p a t h

i n Q . , a n d a n n - c o i l is a l o n g e s t i n d u c e d c y c le . L e t s. a n d c . d e n o t e t h e l e n g th s o f t h e n - s n a k e s

a n d n - c o i l s . T a b l e 2 , c a l c u l a t e d w i t h c o m p u t e r a s s i s t a n c e b y D a v i e s [ 16 ], s h o w s s . a n d c . f o r n ~< 6 ;

t h e v a l u e s f o r l a r g e r n a r e n o t k n o w n . A s s h o w n i n T a b l e 2 , c . + 1 = 2 s . f o r 3 ~< n ~< 5 . D a v i e s

c o n j e c t u r e d t h a t t h i s is t r u e f o r a l l n > / 3 a n d t h a t a n n - c o i l c a n b e f o r m e d b y j o i n i n g t w o

( n - 1 ) - sn a k e s a t t h e i r e n d s . T h e f o l l o w i n g in e q u a l i ti e s [1 8] g i v e t h e k n o w n u p p e r a n d l o w e r b o u n d s

on c . fo r n > t 6 :

7(2 )1) ~< c. ~< 2 - ~ 2n - 12 (1 2)

4(n - 7 n ( n - 1)2 + 2

4 . M A T R I C E S A N D C H A R A C T E R I Z A T I O N S O F Q ,

T h e

a d j a c e n c y m a t r i x A = A ( G )

i s t he p x p m a t r i x i n w hi c h ~u = 1 i f v i i s a d j a c e n t t o v~, a n d

~ u = 0 , o t h e r w i s e . T h e c h a r a c te r is t ic p o l y n o m i a l ~ ( G ) o f G is d e fi n e d a s d e t ( x I - A ) . T h e s p e c t r u m

S ( G ) , is t h e n t h e n o n d e c r e a s i n g s e q u e n c e , A >/2 2 t > . . . t> 2p o f e i g e n v a l u e s o f A [ th e r o o t s o f

~ b ( G ) = 0 ]. F o r e x a m p l e , S ( K 2 ) = ( + 1 , - 1 ) a n d S ( K p ) = [ p - 1 , ( - l y - l ] . T w o g r a p h s G 1, G 2 a r e

c a l l e d

c ospe c t ra l

i f t h e y h a v e t h e s a m e s p e c t r u m , i .e ., t h e s a m e c h a r a c t e r i s ti c p o l y n o m i a l . T h e

s m a l l es t p a i r o f c o s p e c t r a l g r a p h s w e r e g i v e n i n H a r a r y e t al . [ 1 9 ] ; t h e y h a v e j u s t f i v e n o d e s .

C v e t k o v i c [2 0] a n d S c h w e n k [ 21 ] n o t e d t h a t t h e s p e c t r u m o f th e c a r t e s ia n p r o d u c t o f t w o g r a p h s

i s t h e s e t - s u m o f t h e i r s p e c t r a :

w h e r e

a n d

S G t x G 2 ) = S l + S 2 ,

s , = s o , ) = { ~ , , ~ . . . . }

s 2 = S G 2 ) = { U l ~ 2 . . . . }

s , + = + u j} .

A p p l y i n g t h i s r e l a t i o n s h i p t o Q . a s i n e q u a t i o n ( 1 ) , w e o b t a i n :

n

~b (Q . ) = l - I (x - n + 2 0 (7) .

i =

(13)


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A s u r v e y o f t h e t h e o r y o f h y p e r c u b e g r a p h s

1 11 1 11

O I O 1

1 11 1 11

O 1 O I

Fi g 6 The 4 s pa nn i n g t r e e s o f Q 3

283

F o r e x a m p l e ,

~b (Q4) = ( x - 4 )(x - 2)4x6(x + 2)4(x + 4).

A n o t h e r u s e f u l m a t r i x a s s o c i a t e d w i t h G i s it s

connection matrix M.

I t c a n b e o b t a i n e d f r o m A

b y r e p l a c in g th e z e r o i t h d i a g o n a l e n t r y o f - A b y t h e d e g r e e o f th e i t h n o d e . T h e m a t r i x - t re e

t h e o r e m o f K i r c h h o f f [1] s t a te s t h a t i f G h a s t h e c o n n e c t i o n m a t r i x M , t h e n a ll c o f a c t o r s o f M a r e

e q u a l t o t h e n u m b e r o f s p a n n i n g t r e es o f G w h i c h is d e n o t e d b y T(G). S ch w en k [ 2 1 ] ap p l i ed t h i s

t h e o r e m t o g e t

f i (2i ) (7).

(Q,) = 2- (14)

i = 1

F o r ex am p l e , s u b s t i t u t i n g 2 f o r n i n eq u a t i o n ( 1 4 ) , y i e l d s

T(Qz)

= 4 . T h i s i m p l i e s t h a t Q 2 h a s f o u r

s p an n i n g t r e e s a s s h o w n i n F i g . 6 .

T h e d i v e r se a p p l i c a t io n s o f n - c u b e g r a p h s h a v e r e s u l te d i n m a n y w a y s o f c h a r a c t e ri z i n g t h e m .

S o m e r e p r e s e n t a t i v e e x a m p l e s f o l l o w :

( i) F o l d e s [ 2 2 ] s h o w e d t h a t e v e r y c u b e i s a b i p a r t i t e g r a p h s u c h t h a t t h e n u m b e r o f

s h o r t e s t p a t h s b e t w e e n a n y t w o n o d e s x , y i s th e f a c t o r i a l o f th e d i s t a n c e b e t w e e n

them, i . e . ,

d(x,y)

( ii) G a r ey a n d G r a h a m [ 2 3 ] g av e a c r i t e r i o n s im i l a r to ex am p l e (i ), n am e l y , t h a t a cu b e

i s a b i p a r t i t e g r a p h s u c h t h a t t h e n u m b e r o f n o d e - d i s jo i n t p a t h s b e t w e e n a n y t w o

n o d e s x , y o f t h e g r a p h is d(x,y).

( iii ) L a b o r d e an d H e b b a r e [ 2 4 ] f o u n d t h e f o l l o w i n g : l e t C 4 b e t h e c la s s o f co n n ec t e d

g r ap h s s u c h t h a t e ach p a i r o f ad j ac en t ed g es l ie s i n ex ac t l y o n e 4 - cy c le . T h e n w e can

ch a r ac t e r i z e t h e n - cu b e a s f o l lo w s : a g r ap h G i n C a i s an n - cu b e i f an d o n l y i f i ts

m i n i m u m d eg r ee 6 s a t is f ie s p = 2 ~ ) .

( iv ) T h e f o l l o w i n g r e s u l t i s d u e t o v an d en C r u y c e [ 2 5] . A n i n d u c ed s u b g r ap h H o f G is

convex

[2 6] i f f o r a n y t w o n o d e s o f H , e v e r y g e o d e s ic j o i n i n g t h e m i s i n H . A c o n v e x

s u b g r ap h i s proper i f i t is n o t K , , K z , o r G . T h e s e t o f a l l p a i r w i s e n o n i s o m o r p h i c

p r o p e r c o n v e x s u b g r a p h s o f a g r a p h G is d e n o t e d b y P C ( G ) . F o r a n y n > i 3 ,

P C ( Q , )

= { Q 2

Q , _ , }. I f G i s a co n n ec t e d g r ap h s u ch t h a t

P C ( G ) = { Q z, • • . , Q , - ~ } w i t h n ~> 3 an d p = 2 , t h en G i s i s o m o r p h i c t o Q , .

5. E M B E D D I N G A N D P A C K I N G P R O B L E M S

E m b e d d i n g p r o b l e m s a r e c o n c e r n e d w i t h f in d i n g m a p p i n g s b e t w e e n t w o g r a p h s t h a t p r e se r v e

c e r t a in t o p o l o g i c a l p r o p e rt i e s. V a r i o u s e m b e d d i n g p r o b l e m s o f n - c u b e s h a v e a p p l i c a ti o n s i n c o d i n g

t h e o r y [ 1 7 ], l i n g u i st i c s [ 2 7 ] an d co m p u t e r s y s t em d es i g n [ 2 8 ] . F o l l o w i n g R e f . [ 1 ], a homomorphism

h o f G i n t o G ' c a n b e c o n s i d e r e d a s a f u n c t i o n f r o m V(G) i n t o V ( G ' ) s u c h t h a t i f u a n d v a re

a d j a c e n t i n G , t h e n

h(u)

a n d

h(v)

a r e a d j a c e n t i n G ' .

S ev e r a l c la s s e s o f em b ed d i n g a r e d i s cu s s ed h e r e . I f t h e r e i s an

isomorphic embedding

o f G i n t o

Q , , th e n G i s i s o m o r p h i c t o a s u b g r a p h o f Q ,. G r a p h s t h a t c a n b e i s o m o r p h i c a l l y e m b e d d e d i n


8/13

284 FP ANK H~aJat.y e t a l .

an n-cube are called

cubical.

A graph H has an

isometric embedding

into a graph G if and only

if H = G and for all u, v e

V H ) ,

dn u, v) = dG u, v ).

A topological

or

homeomorphic embedding

can be derived from an isomorphic embedding by

subdividing some edges of G so that there exists an isomorphic embedding from the edge-

subdivided graph of G into Q,. The composition of an edge-subdivision and an isomorphic

embedding is then an homeomorphic embedding of G into Q,. Harary [29] defined tcd(G) for an

arbitrary graph G, not necessarily cubical, as the minimum n such that some subdivision H of G

is contained in Q,.

There is another class of embeddings where the range is restricted to subcubes instead of nodes,

and which have a broader definition of distance. A subcube of G can be represented by a vector

X = xl . . . x i . . . x,, where xie 0, 1,, and • denotes a coordinate value that is either 0 or 1. For

example, X = 01. . represents the subcube of Q4 with the node set {0100, 0101, 0110, 0111}. Given

two subcubes

.~ : X I . . . x i . . . X n

Y = Y t • . . Y r . . . Y , ,

the distance

D i X , Y )

between X and Y along the ith dimension is 1 if {x~, y~} = {0, 1}; otherwise,

it is O. Then the

distance

between two subeubes X, Y is given by:

O X , Y ) = ~ D , X , Y ) = ~ d x i, y ,) .

i~l i i l

We say that X and Y are

adjacent

if D (X, Y) = 1. A

squashed-cube embedding,

a concept due to

Graham and Pollak [30], is a one-to-one homomorphi sm from V (G) into a set of mutually disjoint

subcubes which preserves distance as defined above. Figure 7 demonstrates a squashed-cube

embedding of/(-4 onto Q3.

Several interesting problems arise from the various kinds of embeddings. Le t fbe any embedding

function of one of the four types, i.e., isomorphic, isometric, homeomorphic, squashed-cube.

Problem 1

Characterize the graphs which can be embedded in Q, by f.

Problem 2

What is the smallest n for which f embeds G in Q,?

Problem 3

If embeds G in Q,, then for k >/n, what is the maximum number of node- or edge-disjoint copies

of G that can be embedded in Qk?

K,

Fig. 7. A squashed-cube embedding of K4 onto Q3.


9/13

A s u rvey o f the theo ry o f hype rcube g raphs 285

Problem 4

I f f e m b e d s G i n Q , , t h e n i n h o w m a n y w a y s c a n t h i s b e d o n e ?

W e n o w b r i ef ly d is cu s s e ach o f th e se p ro b l em s . H a r t m a n [3 1 ] an d W i n k l e r [3 2 ] p ro v e t h a t w e

can a l w ay s f i n d a h o m eo m o rp h i c o r sq u ash ed -cu b e e m b ed d i n g f ro m K , + ~ n t o (2,. H en ce t h e re a r e

h o m e o m o r p h i c a n d s q u a s h e d - c u b e e m b e d d i n g s f ro m a n y c o n n e c t e d g ra p h o f o r d e r n + 1 i n to Q , .

D j o k o v i c [3 3 ] ch a rac t e r iz ed t h e g rap h s t h a t a r e i so m e t r i c a ll y em b ed d ab l e i n Q , a s fo l lo w s .

Ob vious ly , such g raphs m ust be b ipar t i t e . Le t L (v~ , v2) be the se t o f a ll nod es x of G such tha t

d(v~, x ) < d(v2, x) wi th L s t anding fo r l ess t han . Fo r suf fi c ien t ly l a rge n , a con nec te d b ipar t i t e

grap h G has an i so me t r i c em bed ding in to Q, i f and only i f fo r every edge (vj , v2) of G and a ll

x , y , z ~ L(v t , v:) , d ( x , y ) + d ( y , z ) = d ( x , z ) impl ies y ~ C(v~, v2). I t i s interest ing to no te th at a

cu b i ca l g r ap h n eed n o t h av e an i so m e t r i c em b ed d i n g i n t o an y Q , . F i g u re 8 sh o w s a ex am p l e o f

a cubica l g raph P (3 , 3, 3 , 3 ) which i s i somo rphica l ly e m bed dab le i n Q5 as i nd ica t ed by the l abe ling .

H o w ev e r , i t i s n o t i so m e t r i c a l l y em b ed d ab l e s i n ce d (v , v ' ) = 3 , w h e r e a s d ( f ( v ) , f ( v ) ) = 1.

A l t h o u g h ch a rac t e r i z a t io n s h av e b een fo u n d o f th e g rap h s fo r w h i ch an i so m e t r i c, h o m eo -

m o rp h i c , o r sq u ash ed -cu b e em b ed d i n g i n t o Q , ex i s t s , n o c r i t e r i o n fo r cu b i ca l g r ap h s i s k n o w n a s

y e t . T h e fo l lo w i n g r e su lt s p e r ta i n in g t o P ro b l em 1 h av e b een fo u n d fo r i so m o rp h i c em b ed d i n g s :

(1) I f a g raph i s cubica l t hen i t is b ipar t it e , bu t t he conv erse i s no t t rue [27] . The

smal lest counterexample i s K2,3.

(2) Al l t rees a re c ubica l [27] . The pr oo f by induct ion i s t r iv i a l.

( 3) T w o -d i m en s i o n a l m esh es an d h ex ag o n a l g r ap h s a r e cu b i ca l [34 ].

(4) On e- l egge d ca t e rp i l l a rs span the hyp ercu be [35] .

H av e l an d M o ra v ek [3 4 ] fo u n d a c r i t e r io n fo r a g r ap h G t o b e cu b i ca l w h i ch i s b a sed o n a

t echnique ca l l ed c -va lua t ion for l abe l l i ng the edges of G. A c-valuat ion of a b ipar t i t e g raph G is

a l abe l ing of E (G) such tha t : ( i) fo r each cy c le i n G , a ll d i s ti nc t edge l abe l s occ ur an e ven nu m ber

of times ; (i i) fo r each pa th in G , t here ex i s t s a t l eas t one edg e labe l which occu rs an od d nu m be r

o f t im es . T h e d i m en s i o n o f a c -v a l u a t i o n is th e n u m b er o f ed g e l ab e l s u sed . I t is sh o w n i n [34 ] t h a t

a g r ap h G is cu b i ca l an d G c Q k i f an d o n l y i f t h e re ex is ts a c -v a l u a t i o n o f G o f d i m en s i o n k .

In t u it iv e l y , t h e l ab e ls o n t h e ed g es a r e co o rd i n a t ed w i t h t h e d i r ec t io n s o f t h e ed g es in a k - cu b e

e m b e d d i n g o f G .

T h e u se o f c -v a l u a t i o n s fo r d ec id i n g th e em b ed d ab i l i ty o f h ex ag o n a l g r ap h s a n d t w o -d i m en s i o n a l

m esh es is d em o n s t r a t ed i n F ig . 9 . W e a lso o b se rv e a t o n ce t h a t t h e g rap h p ro d u c t o p e ra t i o n

preserves i so mo rphic em bed dab i l i t y , t ha t i s, i f G~ and G2 are cubica l , t hen Gt x G2 i s cubica l . An

n -d i m en s io n a l m esh o r g r i d is an n - fo l d ca r t e s i an p ro d u c t o f p a th s . I t f o l lo w s f ro m t h e fo reg o i n g

re su lt s t h a t ev e ry n -d i m en s i o n a l m esh i s cu b i ca l. S i n ce m an y p ro b l em s i n s c ien ti fi c co m p u t a t i o n

a re d e f i n ed o n n -d i m en s i o n a l m esh es , t h i s i m p o r t an t r e su l t i m p l i e s t h a t h y p e rcu b e - s t ru c t u red

co m p u t e r s a r e p e r f ec t l y su i t ed t o su ch p ro b l em s .

G a r e y a n d G r a h a m [ 2 3] p r o p o s e d a n o t h e r w a y o f ta c k li n g P r o b l e m 1 i n te r m s o f g r a p h s t h a t

a re no t cubica l . A graph G i s cube-cri t ical i f i t is n o t cu b i ca l an d ev e ry p ro p e r su b g rap h H o f G

i s cubica l . (Thus , such a graph i s min imal noncubica l . ) F igure 10 shows severa l examples of

cube-cr i t i ca l g raphs . Thus , a g rap h i s cubica l i f and only i f i t con ta ins n o cube-cr i t i ca l subgraph .

G a rey an d G r ah a m [2 3 ] an d G o rb a t o v [3 6 ] h av e g iv en p ro ced u re s fo r co n s t ru c t i n g cu b e -c r it ic a l

g rap h s f ro m o t h e r cu b e -c r i t i c a l g r ap h s .

ooo_ ol 10001A

00000 ~ ~ 10000

~ 0 0 l y

1 11

Fig 8 A cub ica l g raph wi th no i s ome t r ic embedd ing


10/13

286 FRANK HARARYet al.

d

4 4 4 4

3 1 5

2 2 2 2

3 1 ~ 5

6 6 6 6

3 1 . J 5

a )

b )

Fig. 9. c Valuations of a 2 dimensional mesh

and a hexagonal graph.

w W

C 3 C 5

K 2 3

Fig. 10. Fou r cub e-critical graph s.

N e x t w e c o n s id e r P r o b l e m 2 . T h e m i n i m u m n r eq u i re d f o r a n i s o m o r p h i c e m b e d d i n g f r o m G

i n t o Q , i s d e f i n e d a s t h e c u b i c a l d i m e n s i o n o f G a n d i s d e n o t e d b y c d ( G ) . T h e f o l l o w i n g s i m p l e r e s u l t

r e l a t es t h e c u b i c a l d i m e n s i o n o f t h e c a r t e s i a n p r o d u c t G t x G 2 t o t h o s e o f G t a n d (3 2; w e o m i t i t s

s t r a i g h t f o r w a r d p r o o f . I f c d ( G , ) = k~ a n d c d ( G 2 ) = k 2, t h e n c d ( G t x G 2 ) = k~ + k 2 . A n i m m e d i a t e

c o r o l l a r y o f t h i s r e s u l t g i v es t h e c u b i c a l d i m e n s i o n o f t h e m e s h G r,~ = P r x P s ( a n d e x t e n d s a t o n c e

t o h i g h e r d i m e n s i o n a l m e s h e s ) :

cd (G, ,s ) = I -log2 r - ] + [ ' log 2 s -]. (15 )

H a v e l a n d L i e b l [3 5, 3 7 - 3 9 ] a n d N e b e s k y [ 4 0, 4 1 ] o b t a i n e d s e v e r a l r e s u l ts c o n c e r n i n g P r o b l e m

2 in t e r m s o f c - v a l u a t i o n . T h e m o r e i m p o r t a n t o f th e s e r e s u l ts a r e n o w s u m m a r i z e d : i f T t2) i s t h e

c o m p l e t e b i n a r y t re e o f h e i g h t n , th e n

c d ( T ~2)) = n + 2 (1 6)

T o p r o v e e q u a t i o n ( 1 6 ) , w e n o t e t h a t a c - v a l u a t i o n o f o r d e r n + 2 e x i st s f o r T ~2). A s s u m e t h a t t h e r e

i s a n i s o m o r p h i c e m b e d d i n g F f r o m T t~ i n t o Q , + t, a n d t h a t a n o d e v i n Q , + ~ i s t h e i m a g e o f t h e

r o o t o f T~fl . T h e n t h e r e a r e 2 n o d e s D o i n Q , + l w h i c h a r e a t a n o d d d i s t a n c e f r o m v , w i t h t h e

s a m e n u m b e r a t a n e v e n d i s t a n c e f r o m v . S i n c e T ~ ) h a s 2 ÷ ~ - 1 n o d e s , o n l y o n e n o d e v * i n Q , +

is n o t i n th e i m a g e o f t h e t re e . T h e n u m b e r o f n o d e s w h o s e d i s t a n c e f r o m t h e r o o t F - J v ) o f T~ ~

i s o d d i s 2 + 2 3 + 2 5--t . . . . w h i c h i s ( 2 + 2 - 2 ) /3 w h e n n i s o d d a n d ( 2 + j - 2 ) /3 w h e n n is e v e n .

N o m a t t e r w h e t h e r v * i s a s s i g n e d to D o o r n o t , t h e n u m b e r o f n o d e s w h i c h a r e a t o d d d i s t an c e

f r o m v c a n n o t b e 2 . H e n c e , T ~ ) c a n n o t b e e m b e d d e d i n Q ~ + ~, s o i t s c u b i c a l d i m e n s i o n i s n + 2 .

F i g u r e 1 1 s h o w s a c - v a l u a t i o n o f T ~2). T h e c u b i c a l d i m e n s i o n o f a c o m p l e t e k - a r y t r e e T~, ~ f o r

k f > 3 , i s s ti ll u n k n o w n . H o w e v e r , H a v e l a n d L i e b l [3 9] h a v e f o u n d s o m e r e s u l ts a b o u t t h e c u b i c a l

d i m e n s i o n o f a c o m p l e t e k - a r y t r e e w i t h t w o l ev e ls , i. e. , T ~ ).

B y a s i m i la r m e t h o d , t h a t i s, b y d e t e r m i n i n g a n n - d i m e n s i o n a l c - v a l u a t i o n a n d s h o w i n g t h a t n o

l o w e r - d i m e n s i o n a l c - v a l u a t i o n e x i s t s , t h e f o l l o w i n g c u b i c a l d i m e n s i o n s w e r e c a l c u l a t e d . N e b e s k y

[4 0] d e fi n e d D f l , f o r n i> 1 , a s t h e t r e e o b t a i n e d b y j o i n i n g t h e r o o t s o f t w o d i s j o i n t c o p i e s o f T ~~)

w i t h a n e w ed g e a n d p r o v e d t h a t

c d ( D f l ) = n + 2 . ( 1 7 )


11/13

A survey of the theory o f hypercube graphs

Fig. 11. A c-valua tion of the binary tr ee T~ ).

287

F i g u r e 1 2 s h o w s D 3~ . S i m i l a r ly , H a v e l [ 39 ] d e f i n e d B ~ a s t h e t r e e o b t a i n e d f r o m T ~ ) b y a d d i n g

o n e e d g e t o th e r o o t a n d s p l it ti n g e a c h n o d e i n t o k n o d e s . H e t h e n d e t e r m i n e d t h e c u b i ca l d i m e n s i o n

o f B ~ ) to b e

cd B ~ )) = n + [_log2 k_J. 18 )

A k - s t a r i s a s t a r w i t h k e n d n o d e s , i . e ., K l .k . A k-quasistar d e n o t e d S h , t2 . . . . t k) is a s ta r l i k e

t r e e t h a t i s h o m e o m o r p h i c t o a k - s t a r i n w h i c h e a c h e d g e e i h a s b e e n e x p a n d e d t o a p a t h o f le n g t h

t ~ . A b i pa r t i t e g ra ph i s

equitable

i f i t c a n b e c o l o r e d b y t w o c o l o r s i n s u c h a w a y t h a t t h e r e a r e

e q u a l n u m b e r s o f n o d e s o f e a c h c o l o r . N e b e s k y [ 4 1] f o u n d t h a t i f S is a n e q u i t a b l e 3 - q u a s i s ta r w i t h

I V S ) l = 2 n f o r s o m e n I> 3 , t h e n S i s a s p a n n i n g t r e e o f Q n , s o c d S ) = n .

F i g u r e 1 3 s h o w s t h e o n l y t w o s p a n n i n g 3 - q u a s i s t a r s w i t h 8 n o d e s . T h i s r e s u l t h a s b e e n

g e n e r a l i ze d t o a c h a r a c t e r i z a t i o n o f t h o s e s t a r li k e tr e e s t h a t s p a n a h y p e r c u b e b y H a r a r y a n d

L e w i n t e r [ 4 2 ] , n a m e l y , t h e e q u i t a b l e o n e s .

P r o b l e m 3 is a n i n s t a n c e o f th e g e n e r a l g r a p h p a c k i n g p r o b l e m i n Q n. F i r s t w e c o n s i d e r t h e

p r o b l e m o f p a c k in g s u b c u b e s i n t o Q n . I f

N Qm c Q~)

is t h e n u m b e r o f d i s t in c t l a b e le d ) m - c u b e s

in Qn for m ~< n, t he n

9

M o r e s p e c i f i c a l l y , t h e node-disjoint o r edge-disjoint) packing, d e n o t e d p ac 0 Q m c Q , ) [ o r

p a c l Q m c Q ~ )], is t h e m a x i m u m n u m b e r o f m - d i m e n s i o n a l s u b c u b e s t h a t c a n b e e m b e d d e d i n Qn

w i t h o u t o v e r l a p p i n g n o d e s o r e d g e s ) . F o r e x a m p l e , p a c0 Q 2 c Q 4 ) = 4 a s i l l u s t r a te d in F i g . 1 4, a n d

pa c l Q2 c Q4) = 8 .

A n i n t e r e s ti n g r e l a te d c o n c e p t is t h a t o f m i s p a c k in g . T h e node-disjoint o r edge-disjoint)

mispacking, d e n o t e d a s m is p a c o Q , , c Q ~) [ o r m i s p a ct Q m c Q , )] , is t h e m i n i m u m n u m b e r o f c o p i e s

o f Q m i n a m a x i m a l n o d e - d i s j o i n t s e t e m b e d d a b l e i n t o Q n- F o r e x a m p l e ,

mi spa c 0 Q2 c Q4) = m i spa c l Q2 c Q4) = 3 ;

s ee F ig . 1 5. T h e w e l l - k n o w n p r o b l e m o f f in d i n g a m a t c h i n g f o r a g r a p h G is e x a c t l y t h e p r o b l e m

o f d e t e r m i n i n g a n o d e - d i s j o i n t p a c k i n g o f ed g e s in t o G . T h e m a x i m u m m a t c h i n g s i ze is

p a c 0 Q i c Q ~ ) a n d t h e m i n i m u m m a x i m a l m a t c h i n g s iz e is m i sp ac 0 Q ~ ~ Q , ) . O b v i o u s l y ,

F i g 1 2 T h e tr e e D f l o f c u b i c a l d i m e n s i o n 5

Fig. 13. Two sp anning 3-quasistars heav y lines) of Q3.


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288

Fig. 14. A node-dis joint packing of Q2 onto Q4.

FRANK HARARY

et al .

r

Fig. 15. Node- and edge-dis joint mispacking of Q2 into Q4.

p a c 0 ( Q l c Q n ) = f l l ( Q n )

a n d m i sp a c 0 Q t c Q ~ ) = f l/ - Q ~ ) ; F ig . 2 s h o w s m i n i m u m m a x i m a l m a t c h -

i n g s f o r Q 3 a n d

Q4

F o r c a d e [4 3] h a s d e t e r m i n e d t h e l i m i t o f t h e r a t i o o f m i s p a c o Q l ~ Q n ) t o

I V Q n ) l w h e n n a p p r o a c h e s i n f in i ty , v i z .,

m i s p a c 0 Q I c Q , ) 1 2 0 )

l im 2~ = ~ .

n ~ c t

A c k n o w l e d g e m e n t s - - T h i s work was s uppor t ed in pa r t by the Off ice o f Nava l R es ea rch unde r C o n t rac t N o . N0014 85 K

0531.

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