Subcube Coverings of Random Spanning Subgraphs of the n-Cube

Math. Nachr. 120 (1985) 327-345

Subcube Coverings of Random Spanning Subgraphs of the n-Cube

By KARL WEBER of Rostock

(Received December 22, 1982)

Abstract. Let G(n, p ) denote the probability space consisting of all spanning subgraphs g of the n-cube En, and the probability is defined as P(g) = p M (1 -p)n3n-i - M if g contains exactly H edges, 0 S X S ~ ~ " - ~ . ERDOS and SPENCER investigated the connectedness of such random graphs for fixed probability p , O<p<l (cf. [l]). In this paper we study coverings of the vertex set of g EG(n, p ) by subcubes of En being also subgraphs of g. Note the analogy between such coverings and coverings of the vertex set of spanning subgraphs of the complete graph K , by cliques.

1. Introduction

The n-cube E" is a graph having the 2n vertices (ai , ..., un), ai=O or I, and two vertices are adjacent if they differ in exactly one coordinate. By choosing the edges of F independently and with the same probability p we arrive at a random spanning subgraph g of E" with probability P(g) = p M f - M , q= 1 - p , N =n2?'-', if g has exactly fl!f edges, 0 SLTI s N . All these random graphs form the probability space G(n, p ) . We say that almost every (u.e.) graph g€G(n, p ) has the property Q if P(g has &) -1 as n +OD. For example, if the probability p is fixed and p =- 1/2 (p-= 1/2), then a.e. g€G(n, p ) is connected (not connected) ; if p = 1/2, then P (g is connected) -.lie as n -.CO (cf. [l]).

A covering C of gsE" is a covering of the vertex set of g (i.e. of the vertex set of E") by subcubes of En being subgraphs of g too. The number Z(C) of subcubes in a covering C is called the length of C. A minimal covering C of g has minimum length among all coverings of g. This minimum is called the length of g and is denoted by Z(g). A subcube K s g is said to be a maximal subcube of g if there is no subcube K' with K c K S g . The number of maximal subcubes of g is denoted

An irredundant covering of g contains only maximal subcubes of g, and non of them is redundant. I t is clear that Z(g) =min Z(C) taken over all irredundant coverings C of g. Define Z+(g) =max Z(C) taken over all irredundant coverings C of g and denote by t(g) the number of all irredundant coverings of g.

To construct minimal coverings for a given g there is a unifying approach: First determine all maximal subcubes of g, then construct all irredundant coverings of g and finally find the minimal coverings of g among the irredundant ones.

by 4 g ) -

328 Nath. Nachr. 120 (1985)

The complexity of this process may be characterized by parameters as s (g ) , b(g) or z(g). If we renounce the determination of minimal coverings and replace it by an arbitrary irredundant one, the ratio Z+(g)/Z(g) gives the possible deviation of the minimum length.

Obviously the parameters introduced above are random variables on G(n, p ) . In nest sections we shall bound it for a.e. graph gEG(n, p ) . To do this we use ideas and constructions of A. A. SAPOSHENEO (cf. [Z] -[5]) and their generalizations to random BooLEan functions in [6] -[lo].

We use the following notations too. (All limits, asymptotics etc. are considered as n+-.) We write a(n)s@(n) (a(n)zp(n)) if a(%) sp(n) (I + o ( i ) ) (a(n)=O(g(n))), ct(n)-/l(n) if a(n)/@(n) -1 (cc(n) and p(n) are asymptotically equal) and a ( n ) x b ( n ) if x(n)z / l (n ) and @(n)sa(n) ( ~ ( n ) and P(n) are of the same order). Everywhere q~ = y ( n ) denotes a function of n which increases arbitrarily slowly to infinity. The natural and the dual logarithm are denoted by In n and log n, respectively. For any real 2, 1x1 and Tzl denote the greatest integer not greater than x and the least integer not less than z, respectively.

The distance 6(g, _b) between two vertices a , &E" is the customary HAMMING distance, i.e. the number of coordinates in which a and _b differ. The distance between two subcubes K and K'of P i s defined as d(K, K ' ) =min d(@, b ) , where the minimum is taken over all acK and _bcK'. For a fixed d-cube KSE" there are exactly (n - d ) d-cubes K' SIP which form together with K a (d + 1)-cube. (Note that vertices and edges are 0- and l-cubes, respectively.) These subcubes are called neighbouring or the neighbours of K . Clearly K g g is a maximal subcube of g iff non of the n -d neighbours K of K together with the 2d edges between K and K' belongs to g. A d-cube K s g is called an isolated d-cube of g if g does not contain an edge between K and IT-R. Denote by i d ( g ) , j,(g) and sd(g) the number of d-cubes, isolated d-cubes and maximal d-cubes of g, respectively. These parameters are random variables on G ( n , p ) too. Write i d , j a and for the corresponding expectations. Then the following assertion is evident :

d-1 Proposition 1.1. Setting y=p(d+~) '~- ' and z=pd2 we have

(1.1) id=T * Z ,

(1 2) j d = T . . ,cn-azd

(1.3) d=O, 1, 2, ...

and D2X the variance of X. Let t w o , then

j d=T * 2 * (1 - y ) " - d ,

Let X be a non-negative integer-valued random variable, E X the expectation

(1.4) P (Y Z t Z X ) s l / t (inequality of M ~ x o v ) . It implies

Proposition 1.2. (1.5) P ( X S q I , E Y ) - l ,

Weber, Subcube Coverings 329

and, in particular, (1.6) P (X=O) +l if EX - 0 .

The following inequality of CHEBYSHEV is an immediate consequence of (1.4) :

(1.7) P ( jX -EX/ Z t ) s DzX/t’ . This implies

Proposition 1.3. If (1.8) E X - m and DzX=o((EX)Z) , then

(1.9) P ( X - E X ) +l ,

and, in particular,

(i.e. P(IX-EXI<&EX)- l for a certain E = E ( ~ ) - + O )

(1.10) P , ( X Z l ) - l . Note that (1.8) can be replaced by

(1.11) EX +- and E(X),- (EX)Z as DZX = EX?. - (EX)2 and E ( X ) , = EX (X - 1) = EX2 -EX is the second factorial moment of X .

Let xd denote one of the random variables id, jd or 8d and given the integer sequences d’=d’(n) and d” =d”(n), 0 s d ‘ ~ d “ . Then using (1.7),it is not difficult to prove the following

Proposition 1.4 ([9]). If

then for a.e. graph gcG(n, p ) we have (1.13) Xd(g)-EX,j for all d With d ’ s d s$’

Note that (1.13) implies d” d’ ’

(1.14) 2 x d ( f ) - * d=d‘ d a d ’

2. Subcubes

We start with estimates for the variances D2id and D2sd. Lemma 2.1. Given the integer d 21 and the probability p , O-=p i: 1. Then

330 Math. Nachr. 120 (1985)

Proof . By standard techniques we obtain

where (") (") (" -') Zn-' is the number of ordered pairs ( K , R') of d-cubes d s d - s

which have exactly an s-cube in common and pd2d-s28-1 is the probability that both cubes K and K of such a fixed pair are subcubes of gEG(n, p ) . (If d = 1, then one has to replace the sum in (2.1) by 0.) If s = 0 or K and K have no common vertex, then P ( K UK' 2.9) =pdZd and consequently the sum over all these ordered

Lemma 2.2. Given the ,integer d z I and the probability p , 0 -=p -= 1. Then

T h e bound (2.3) remains true for d = o if we replace the first ~ ~ L ~ ~ ~ n d in the brackets bg 0.

Proof. Denote the d-cubes of F by K I , K,, ..., K,, T = ( i ) 2h -a , and define

the random variables Xf, ..., XT on G(n, p ) as follows:

1 if K, is a maximal subcube of 9, XU(9) = { 0 otherwise.

Then we have E(sd)i=

ting z = p

EX,X,. If the distance 6(Ku, K,) is at least 3, then set- u+u

dzd-1 we obtain E-X,X, = ( z (1 - ~ ) ~ - ~ ) 2 , so that

c EX,X,sT*(z ( 1 -y)"-d)Z=S; d(K,,K,) r 3

(cf. (1.3)). If 6(Ku, K,) = 1 or 2, then there are 2 (n -d -1) pairwise disjoint neighbouring d-cubes of K , and K,, respectively, what implies Ex,X, s z z (1 - y)2'n-d-1)

Snd because there are at most T2d (i) (n + (3) = T20 ($) such ordered

pairs (Ku, K,), we deduce

C EX,Xu = O( A d) Sj . 1 Sd(K,, E,)SZ

Note that d=O and K,+=K,implies 6(K,, RJz1 such that E(so)2~(1+0(oo)) 3,". Suppose now that 6(K,, K,) = 0 and let the d-cubes K, and K, have exactly an

s-cube in common. Then the n - 2d + s neighbours of K, and the n - 2d + s neigh-

Weber, Subcube Coverings 33 1

bows of K,, which do not intersect Ku a,nd K,,, respectively, form n-2d+s pairwise disjoint copies (Ki , Ki ) , ... 9 2 1 Kn--ld+' ) of the pair (K,, K,,), i.e. RLnK: is an s-cube too for all r = 1, ..., n - 2 d f . s . Assuming the maximality of I<, and'K,,, all the d-cubes KL and KZ, T = 1, ..., n -2d +s, do not belong to g. The probability that such a fixed pair (KI , Ki) does not belong to g is

) 2 1 -p(8 +2)28-1 + p ( ~ + ?)38-1 ( 1 -p(d +?)Zd

='1-2y+p (3d + 7)ad --?

- ( 8 + 2 )?'- I

1 - 2y + y3/?,

so that E X , , X , S Z ~ ~ - ' ~ ~ - ~ (1 -2y+y3'2)n-'d. Since the number of pairs (K,,, K u ) ,

. We --sp - 1 - (a - - 2 Recall that d - i (s) ( d - s ) % ( ; ) n-d and bound 2 - ' 1 1 and p Z p s=o

complete the proof by noting that D%,=E(-s~)? + sd -ii. Q.E.D.

Remark. The function h(z) decreases monotonously in the interval [ O , 1) and

In the following theorem we give thresholds (cf. [l]) for small (maximal, l s h ( x ) > l / 2 for Osx-=l .

isolated) subcubes.

Theorem 2.3. Let k be an arbitrary fixed non-negative integer and put

k I n n cp ' l l ( k + 2 ) Z k - l p.=2- - --) . Replacing the term - 97 (-;) in p , n

(in p i ) by -- - we get the probability p3(p i ) . Then a. e . graph gEG(n, p ) n n ("1 (i) contains no (maximal, isolated) k-cube i f p =o(p,), k zl, and (ii) contains asymptotically ik k-cubes if p i p L -fa, k L 1, and (iii) contains asymptotically jk isolated k-cubes if p/p1--, k 21, and p s p 2 and

(iv) contains asymptotically i k maximal k-cubes if pip l -03, k z 1, and p z p i and

Proof. Recall that ik= (:) 2n-kpk'k-1=(p/pl)82P-1 -0 if p=o(p l ) . Thus, (i)

follows from (1.6). Further we see that ik+m if p/pl--. Using the bound (2.1) it is easily shown that D2i, = o( ii) in this case. Hence in accordance with Proposition 1.3 assertion (ii) is derived.

does not contain an isolated k-cube for p z p 3 and

does not contain a maximal k-cube for p %pi .

332 Nath. Nachr. 120 (1985)

In order to prove (iii) k t us remark that j k -+a for all p satisfying. ypi s p Sp?.

takes on its minimal value in the given interval at the boundary because the deriva- tive is monotonous. Now we show (1.11) for X = j k . Denote the k-cubes by Kl, ._., KT again and define the following auxiliary random variables on G(n, p) :

Indeed thisiseasily checkedforp = yplandp =p2but thefunctionp k2k-i (1 -p)(n-ky'k

1 if K,&g is isolated, 0 otherwise Yu(g) = {

u=l , ..., T . Recall that E ( j k ) , = 2 EY,Yv. Since for a fixed k-cube I i S E n

there are at most 2k (1 +n) = O(nkfi) k-cubes K' with 6(K, K') s 1, there

exist at least T2 (1 -O(n/P)) ordered pairs (Ku, K,) with 6(Ku, K,) ~ 3 , i.e. with EYuYv=(pkzk-l (1 -p)(n-k)ak)2. Obviously EYuY,,= 0 whenever 6(KuJ K,;) = 0. If 6(K,,, K,) = 1, then there are at least 2 (n - k) 2k - 2k forbidden edges in order t o ensure that both K , and K, are isolated k-cubes. Thus we conclude

u 8 V C)

- O ( l ) < -2

Pinally we will prove (iV). Simple calculations show that .?k+O0 for p with ?pi s p s p i (for p Spi if k = 0) and sk -0 for p sp;. Now we use the upper bound

E(jk)2siifO(n/2n) ji (1 -p) - y k .

n3 22k - 0. Suppose k z 1 and put f& = - 2n (1 -p) 3 e k

and no=

(2.3) in order to check (1.8) for X = s k . For k =O and p we have So +a 2

k - 1 k - I -- -n- 2k Assume qp l sp=o( l ) . Then p(k-1)2k-2 w n ' 2 -

and consequently @k --. Considering that h ( f i ) S i k - 1 - c s p s p i , where cis a positive constant. Thenp(k-1)2 -1 and (1 -y)nz2-nn,-o '1)

and thus ekzn-'(') . Furthermore c s p implies c' s G a n d h( vy) S c " i 1, where c' and c" are also positive constants. Therefore Qk+0 in this case too. Our proof is completed by noting that

-

n?

2n n k x - - - o .

Q.E.D.

Remark. Note that s,(g)=j,(g) and that io(g)=2" for all g€G(n,p). By the method of momentsit is not difficult to show that for p-xpl, x+ 0 a real constant, the number of (isolated, maximal) k-cubes is asymptotically Porssow distributed with mean value

xk2k-1 /Ir A=- ~ Z I , i.e. P ( i k = r ) - P ( jk=?-) ' -p ( s k = r ) - - e - " k ! Z k ' r !

r = O , 1, 2, ..., kl .

Weber, Snbcube Coverings 333

For p = p 4 = 1 - 2- we can show that j , is Porssox k k-1 distributed with mean value Y=ez (1 -2- ' - )'' /k!, k z O . In particular, that

and, assuming x = 0, we have 2 2n

fez)' -fl implies P ( jo=r)-- e r !

1 2

P (jO=r)-l/er! and P(jo=O)-P (g is connectedj-lfe for p = - + o (t) (cf. [l]).

Lemma 2.4. Theratio~,=S~+~/~~iiSmonotonouslydecreasingford=0, 1, ..., n - 1. Proof . By (1.3) we have

1 - ypZd Setting ad=- we are home if we show that ad+l %ad. But the last inequal-

ity follows if the polynomial 1 - Y

f ( x ) = - x 4 d t . ~ 6 + x : 3 d + 1 4 + x 3 d + 1 0 - 2 x d + 4 + 1

is positive in the open interval (0 , 1). We have f ( O ) = l and f(l)=O. By the sign rule of DESC-ARTES f(x) has a t most 3 positive roots. Since f '( 1) = 0 and f " ( 1) = 8 > O , f(z) has a local minimum at x= 1, and x = 1 is a double root of f(z). Thus, there is no a€ ( 0 , 1) with f (a) = 0. Q.E.D.

Let the probability p =p(n) be given. The following theorem gives an answer to the question which dimensions of (maximal) subcubes are available for a.e. 9 EG(n, PI.

Theorem 5.5. Given p =p(n) such that 3

- Y; 2 ( 2 . 5 ) 2 spsl---.

log n - - Put do=rlogn-loglogn-log~og (1/p)jandfor,pz1/1/2 d,=r-log Iog(l/p)--log ( -log log (l/p))]. Then a. e. gEG(n, p )

(i) contains asymptotically id d-ctcbes for all dimensions cl= 1, 2 , ..., do and contuins no (maximal) d-cube of dimensions d z d o + 2 and

(ii) contains asymptotically Sa maximal d-cubes for all dimensions c l = 1, ..., cl, if p 5 1 1 2 and

(iii) contains asymptotically 3, maximal d-cubes for all dimensions d = d3 + 2, . . ., do

,and contains no maximal d-cube of dimensions d = 0, 1, . .., d3 ,if p z 1/ i 2 ( = 2- ).

Proof . Set d~=logn-loglogn-loglog (l/p) and d=do+2 . Then i d s

s Tp (d*+Wd'+ '= T~ (cf. (i.1)). Taking account of (2.3) we obtain

(2.6)

512 - 3 - 9

-2n + ' l ~ ( i o g l o g n +loglog(ilp)-Z)

1 3

-log log n + 1 s log log (1/p) 5- log n

334 Math. Nechr. 120 (1985)

and

(2.7) - logn-loglognsd:s logn-1. 2

3

monotonously decreases for d=O, 1 , .... Thus, min id=min (zi, is>, and we

checked that id-- for all d, 1 sdsd , . l s d s &

~

Using doSd:+ 1 Slog n and log log %+log log ( l / p ) nl (cf. (2.7) and (2.6)) we

Now assertion (i) follows in accordance with (1.6) and Proposition 1.4. Suppose p s 1/2 . By Lemma 2.4 the ratio vd = 8d+I/8d is monotonously decreas-

ing in d. Consequently in the interval 1 S d Sd,, the minimal && is 8, or 8&. Easy calculations show that 8&- iG +OO and 3,=n2n-ip (1 - ~ 3 ) ~ - ' ~n2n(i-'og(s'7)-o(1))--

o2d i a, L I

@d sd d = l too. Put e d = - + - + 0 ( a d ) (cf. (2.3)). Then z & d ' O is seen as follows:

.d 9% n2 log n n2 log n s 5-, 22d

do-sdd, d:?dz-i 2n'2(7/8)-R - 1,2n

znp (1 -P3)" Qd

do logn -5- - -0 is clear by the lower bound for 8, (see above), 8 , 8,

n4 log n n4 log n _-- A . Renee, by Pro- proof of (i) for a lower bound of i4), d o a d s 2-R(l-p3)? 2-R

position 1.4, we established (ii). 513 -

Assuming p 2 I/ 1/2 one shows 2 &d -0 in the same way : d = d 3 + 3

gZd n2logn n'logn do-z--5 @d 2n/200,73n - 1,03n

Weber, Subeube Coverings 335

16 log n -5 ---0 d0

(see. above),

( l i ; +3)& -

- -- 2 -lOglOg(l/P) z c > 1 / 2 , and that consequently & + o , the proof Of (iii)

By noting that for d s d 3 one can bound 3, s T (i - p (d;+3)& ) n - d , where p - 1 log(-loglog(l/p))-3 d3

2 d =o is also completed. Q.E.D.

Remark. By Theorem i we h o w that for k = 0, 1, 2, ...

- 512,- implies 8 k -0 andsk+i +-. Easy calculations shorn that for 1/y2 ~p s i/ 1'2 assertion (iii) of Theorem 2 remains true if we put d3= [d;] - 1 in this range of probabilities.

Theorem 2.6. Given p =p(n) such that

Set d?=Llog log n-log log log n-log log ( l / p ) ] . Then a.e. gCG(n, ZJ) (i) contains asymptotically T ~ + ~ + ;&+3 subcubes and (ii) contains asymptotically sd2 + 2 + gd2 + maximal subcubes. P r o o f . Put d:=log log n-log log log n-log log ( l / p ) . An easy calculation

yields

Furthermore, from (2.8) we deduce

(2.9)

and

(2.10)

1 2

-log log log n + 3,Ol s l o g log ( l / p ) I- log log n -log 1% 1%

1 -log log n s d : Slog log n - 3,Oi 2

336 Math. Nachr. 120 (1985)

for sufficiently large n. Considering that p d monotonously decreases in d we find

for certain w = w ( n ) -a. Consequently Td is maximal for d =d9 + 2 or d = d 2 + 3 and d? + 1 do+? &-d,-2

a=o 2 ' d S ' d Z + 9 u=o ( t ) Y C i d 2 + 3 u=o 2 ( t ) u N ' d 2 + ? + T & + 3

and by (1.14) we have (i).

Obviously V d g p d , but for d=d2+2 we get p(d+2)2"-1

this case too. Thus, as above, there exists an w = o(n) -- such that

By Lemma 2.4 the ratio v d = i d + i / i d is monotonously decreasing in d too.

=O - So that V d z p d in (3

de +i Hence sdis masimalford=d2+2 or d = d 9 + 3 and S d N i d 2 + ? + & a f 3 . Q.E.D.

d=O

3. Minimal coverings

In this section we derive some bounds for the length Z ( g ) of a.e. gEG(n, p ) . For d = 1, ..., n and an arbitrary fixed vertex g EE" define the following random variables on G(n, p ) :

V d , a ( g ) =number of d-cubes K g g which contain the vertex a. For the expectation cd = (independent of g )

is easily checked. Since the variance is also independent of n, set D 2 w d = D % d , a . -

dZ=o(n). Then Lemma 3.1. Given p = p ( n ) , O-=p-=I, and the integer s e q z i e n e e d = d ( n ) 31 ,

and ("-") s(") 5 we obtain (3.2). (For more detailedevplanationssee the proof

of Lemma 2.1 and ["I, proof of Lemma 1.) Q.E.D. d - 1 d n


Lemma 3.2. (i) If 6 d - 0 , then for a.e. gCG(n, p ) d-cubes KGg cover 0 ( 2 ~ ) vertices on1 y.

(ii) Let D?vd S &$:, &d = Ed(n) - 0. Then a.e. gcG(n, p ) contains more than (1 - P)&d) an vertices each of them is covered by asymptotically 5, d-cubes K &g (i.e. for arbitrary 9 -coo there exists an E -0 such that a.e. g there are more than (1 - fp&d) 2n vertices a satisfying lVd,a_(g) -5dI s & 2 i d ) .

Proof . (i) By (1.5) for a.e. g d-cubes of g cover at most @idvertices. Assum- ing Gd -0 this is valid, in particular, for functions 9 with ycd -0. But this means

(ii) Define bd(g ) as the number of vertices ~ E E " with lVd,&) -5dl s & * d . Using that f p 2 d c d = 9 f i d 2 n = O ( 2 n ) .

(1.7) we can bound the expectation 6 d =Ebd as follows : 6, = 2nP (for a fixed a

1 / ~ 2 -.to arbitrarily slowly. Our assertion now follows from (1.5). IVd,,&) -ijdl z & i d ) s _ 2 n D ? c d / E 2 5 ~ ~ 2 n & d / & 2 , and We Can choose &=&(n) -0 such that

Q.E.D. Corollary 3.3. Given the integer sequences d' =d'(n) and d" =d"(n ) , 1 s d ' s d " .

Let g d - - and D % d ~ ~ d 5 i , d 'sdsd ' ' , and (a"-&) 2 E d - 0 . Then for a.e.gE

EG(n, p ) asymptotically 2n vertices have the property that each of them will be covered by asymptotically $d d-cubes of g for all dimensions d =d', ..., d" (i.e. there exist E , E' -0 such that a.e. g contains more than 2n (1 - e l ) vertices whichsatisfy / V d d ( g ) !- -

d "

d a d '

-6d1 s & c d for d=d ' , ...) d"). Proof . Put again bd(g) = I{a: Ivd,g(g) -5dj Z E * ~ } I . We proved above that the

expectation 6 d % z n & d / E 2 . Further, by (1.5), P(bd(g) z P ) 6 d ) d l/g, and consequently (u, n and 2 are taken over d = d ' , ..., d") P ( U { b d ( g ) ~ g i g J } ) ~ C F ( b d ( g ) z

d'-d'+l 2 f$d) (&' - d' f 1)/p and P ( n (b&) CI ( P 6 d ) ) 4 1 -

m n { b d ( g ) < @ d )

Y v Eg .-"

we have 2 bd(g) S - (c E d ) 2", and by our assumption we can find & -0 and Y

E - P) -00 such that (d" -d ' ) /p -0 and I (z E ~ ) -0 too. Q.E.D-

Theorem 3.4. Let k z 1 an arbitrary fixed integer and p =p(n ) such that pn112k-1 - -r- and pn'/2k -0. Then for a.e. gEG(n,p) asymptotically Zn vertices have the

property that each of them is covered by asymptotically v d d-cubes of g for all dimensions d = 1, ..., k, and d-cubes of g of dimensions d ~ k + 1 cover only o ( P ) vertices.

by Lemma 3.2 (i) we conclude the second assertion. Let 1 s d s k and put & d =

-01 C P d

nP Ud

(;) pk2k-i.-. , (n l /2k- ;3)k?k- l =-+_ (cf. (3.2)). We have C,=np--r- and c k =

too. Considering that fd+,/f id monotonously decreases we get

22 Xath. Nachr., Bd. 120

335 Math. Nachr. 120 (1985)

&d+O for 1 z d sk and so, by Lemma 3.2 (ii), we established the first statement too. Q.E.D.

Theorem 3.5. Let p =p(n) such that 5

(3.3) n-N1) s p 1 - y-GiG)

and put dl=[log log n-log log ( l / p ) ] . Then for a.e. gEG(n, p ) asymptotically 2n vertices have'the property that each of them is covered by asymptotically 6, d-cubes of g for all dimensions d = l , ..., dl- l , and d-cubes of g of dimensions d z d l + l cover only o(2") vertices.

di - l d3 d Proof . P u t Ed=-+-. We will show that di By Corollary 3.3

nP V d d = l this is sufficient for the first part of Theorem 3.5. The given range of probabilities

implies n p zn,i-o(i) and di= o(6 ) . Zurther for d =d, - 1 we bound Gd zny2 /dd -00

as well as 6Jd:. Put d =di+ 1. Then c d S ( ~ ) p d z d 1 8 ( ~ ) n - d ~ l ~ a ! + o as for p z

- >n-'(') dl+m. By Lemma 3.2 (i) the proof is complete. Q.E.D. From Theorems 3.4 and 3.5 lower bounds for the length of a.e. random span-

ning subgraph of E" can easily be derived. To construct coverings of small length for a.e. g we use two methods of SAPOSHENEO ( (3) , (4)) and their generalizations t o random BooLEanfunctions (cf. (7)). First we calculate the complexity of a greedy algorithm: In every step take such a subcube of g which covers the most of -until to this step -noncovered vertices among all subcubes of g.

Lemma 3.6. ([4]). Given a 0 - l-matrix H of size n x i which has such a (1 - E ) n x x m-submatrix P!. that every row of & ! contains at least v 1's (cf. f i g . 1). Then every greedy covering of M contains at most

5

m ( l n L + i ) + s n + i nu (3.4)

fig. I

columns. ( A set of columns covers a matrix M if every row of H has a common 1 with a chosen column.) How to apply this statement in the case of our covering problem

shows f i g . 2 and the

Weber, Subcube Coverings

' d

339

Corollary 3.7. ([7]). Let g d + r n , D%,S&&, ed+O, id+-, D%d=o(ii). Then foy a.e. gEG(n, p ) the length

(3.5) Z(g) 5 (dln2 + 1) + y ~ ~ 2 ~ . The corollary is an immediate consequence of Proposition 1.3, Lemma 3.2 (ii) and Lemma 3.6. Note that (3.5) implies, in particular,

(3.6) if d=0(1) and (3.7) Z(g) 5 2"-'dZn2 if d +- and &d=o(d@).

pn''2k-i -01 and pn""+O. Then for a.e. g c G ( n , p ) the length (3.8) 2 n - k 5 Z ( g ) 5 2 n - k (kln2+1)

Z(g) 5 2n-d (dZn2 + 1)

Theorem 3.8. (i) Let k z 1 an arbitrary fixed ,integer and p = p ( n ) such that

(ii) Let I ) = p ( n ) such that

and put d,=[log log n- log log (1/1))1. Then the length of a.e. gEG(n, I ) ) satisfies (3.10) 2 n - d i ~ Z ( g ) ~ 2 2 n - d i f i d , l n 2 .

Proof . The lower bounds are clear from Theorems 3.4 and 3.5 which say that we have to cover asymptotically 2" vertices by subcubes of dimension at most k or d,, respectively. The upper bounds are obtained by the greedy algorithm: I t is easy to check that the conditions of Corollary 3.7 are satisfied for d = k or d= =d,- 1, respectively. Q.E.D.

Now let us describe the d-matching -algorithm ([3], [7]). A set of 2n-d pairwise disjoint d-cubes of g which cover 2"-0(2"-~) vertices is called an asymptotical d-matching of the spanning subgraph g. We shall construct an asymptotical d- matching for a.e. gcG(n ,p) . Let xI, z2, ..., xn be the coordinates of the n-cube and let Qi, 1 ~i st = Ln/dj, denote the set of all d-dimensional subcubes of E" with edges in directions ,'..., x i d . The algorithm works as follows. In the first step we take all subcubes of Q1 contained in g ; then we take all subcubes of 22.

340 Math. Nachr. 190 (1985)

Q2 contained in g and without common vertices with already taken subcubes and so on. Let S,(_a) be the probability that a fixed vertex GEE'' is not covered by any chosen subcube after i steps of the prescribed algorithm. Obviously &(a) is the same for all a E E " , therefore we omit the vertex. By the recursion di = SiWL - S;-,z, do= 1, it is not difficult to prove the following upper bound for 8,:

Id

Lemma 3.9 ([3], [7]). R--

( 3 . 1 1 )

where t = Lnld], z = p

S, sl/ /l f z R t , a2d- 1 , R = 2d - 1 .

Lemma 3.10. For a.e. gcG(n, p ) (3.12) Z(g) 2n-d+p3t2" .

Proof . Define the random variables c d ( g ) =number of vertices of g which are not covered by the d-matching algorithm. We evaluate the expectation Fd=2"P (a fixed vertex is not covered)=PS,. Thus, by (1.5), for a.e. g€G(n , p ) c ( g ) s 5 9142'' and consequently ( 3 . 1 2 ) is valid. Q.E.D.

Theorem 3.11. (i) Let k z 1 an arbitrary fixed integer and p = p ( n ) such that pnllkak-' +m. Then for a.e. gcG(n, p ) there exists an asymptotical k-matching, i . e . (3.13) Z ( g ) 5 2n-k .

(ii) Let p = p ( n ) such that (3.14) n--o(lfloglogn) s p s 118

and put d,=[log log n -log log log n -log log ( l / p ) j . Then the length of 8.e. g $ cG(n, p ) satisfies ( 3 . 1 5 ) l ( g ) 5 2 " - d 2 .

= 0 ( 2 - ~ ) for d = k or 6=dz, respectively. By (3.11) Proof . In accordance with Lemma 3.10 we are home if we show that a,=

1 log (8 ,zd) s d -- 2d log zn .

(i) For d =!$ we have log xn = k2k-1 log (pn"k3k-i) and consequently log (?&ak) 5 log (pn"k2k-i) + -m, as claimed.

k =sk-- 2

1 d 2 2

(ii) For d =d, we obtain log z n =- log ( p n ~ld"- ' ) and easy calculations

2 log log n show that in this case log

z( -- i log (lip). The last term

tends to infinity if p -0, i.e. log (S,zd) -+ --, as claimed. Ifpxl , p I 1/8, the same

-i z3 for sufficiently large n, and so

Q.E.D.

term is no smaller than 3 d ( 2 )

log (8,2 ) ~ a - i , 5 d = -o,5d---f0ra2--.

Weber, Subcube Coverings 34 1

We remark that, of course, assuming p=-1/8, for a.e. gEG(n, p ) there is an asymptotical d-matching for d = Llog logn -log log log n -log 31 too, but we can not prove this for d =d2 in this case. We give some examples how the statements (i) of Theorems 3.8 and 3.11 can be applied to obtain bounds for Z(g) for a.e. g€G(n , p ) :

1. p n -m and pn"' -0: Z(g)- 2"-'. The upper bound is obtained by Theorem 3.11. If we onlyassumethat p G=O(l), then we can onlydeduce that Z(g)Nz2n-2.

2. p f n -CQ and pn'14 -0 : 2n-2 5 I (g) 5 Zn-l. The upper bound by Theorem 3.11 is better than Z(g) 5 2n-2 (2 In 2 + 1) by Theorem 3.8. Again pn114 = O( 1) yields only the weaker lower bound Z(g) 2 T 3 .

3. pn'l4 -+a and pnl/' -0 : 2"-3 5 Z(g) Zn-' . The upper bound by Theorem 3.11 is better than Z(g)S2n"-3 ( 3 In 2 + 1) by Theorem 3.8.

4. pn'l' -m and pn1I1' = O( 1) : 2n-4 5 Z(g) 5 2n-4 (4 In 2 + 1). The upper bound by Theorem 3.8 is better than Z ( g ) s 2 n - 3 by Theorem 3.11.

5 . pnl"' -00 and pn1i16 -0 : 2 " - 4 5 Z(g) 5 Z n W 3 . The upper bound by Theorem 3.11 is better than I ( g ) s F 4 (4 * In 2 + 1 ) by Theorem 3.8.

On the other hand the statements (ii) show that for largep the greedy algorithm yields a better upper bound than the d-matching-algorithm :

-

dl 54 log ( U P ) *log. log n log -5- (UP) 1 - -

zd' - log n log n

4. Irredundant coverings

Let F = F ( n , p ' ) be the probability space of all random induced subgraphs of E" (random BooLEanfunctions of n arguments). The probability of such an induced subgraph with exactly m vertices is defined as P ( f ) =p'" ( 1 - p )* . (For investigations of random BoOLEan functions see [6] - [ 101.)

Define the probability space H = H ( n , p , p ' ) =G(n, p ) X F ( n , p ' ) consisting of all pairs h = (g, f ) , g EG and f EF, and the probability of such a pair is defined as Prob (h )=P(g) P(f). As customary Q S H contains a.e. pair if Frob (&) -1. The number 191 of edges (on G(n, p ) ) as wellas the number if1 of vertices (on P(n ,p ' ) ) is binomial distributed, i.e. in particular that for a.e. h = (9, f ) E H we have Is/ - -pn2n-' and I f 1 -p'Zn provided that these values tend to infinity.

, 7"-m

Lemma 4.1. Given A g H with Prob ( A ) -0. Then for a. e. g EG there exists an

Proof . Let A'={g: ( g , f)EA for all f with Ifl-p'2"}. Then we have

f c F , ~ f ~ - p ' Y , such that (g, f) 6A.

Prob ( A ) I Frob ( h ) z P(g) 2 P ( f ) 2 2 W ) = P ( A ' ) , /1=k7.f) !?€A' Ifl-P"" BEA'

BE A'

and thus P(A') -0 too. Q.E.D.

342 Xath. XTachr. 130 (1885)

Exactly the maximal subcubes of g are the maximal subcubes of h=(g, f ) . A subcube K is called an embedded subcube of h = (g, f ) if IK-f I s 1 (difference between the vertex sets). For d z 1 and e y e q vertex cz<P we define

Wdo(g , 1- f ) =number of maximal embeddedd-cubes K of (9, f ) with aER and K G f U a .

These parameters are random variables on H ( n , p , p ' ) . Since the expectation EWd,, is independent of a, we write zd for it. By the same reason we write D'w, for the variance. I t is easily checked that

(4.1) % d = ( i ) d ( l - y ) n - d , d Z I ,

where z=p dZd-l , =p(d+2)2d-1 and z (1 - z J ) " - ~ is the probability that a fixed d-cube K is a maximal subcube of h = (g, f ) . The value z' =p'"-' is the probability that K 5 f U a (a E En arbitrarily fixed).

Lemma 4.2. Given the probabilities p and p' with pnzn -03 and ~ ' 2 ~ -w and the integers d = d ( n ) '1 1. Suppose that

(4.2) %d +m and D2Wdz&d%: , where & d = & d ( n ) -0. Then for a.e. pair he H there are at least (1 - y & d ) 2% vertices which are covered by asymptotically c d maximal embedded d-cubes of h.

Proof. For a pair h € H denoteB(h)=(a: IWd,a(h)-GdlZ&?&) and b(h)=IB(h)l. Then the proof is exactly the same as the proof of Lemma 3.2 (ii).

Q.E.D. Corollary 4.3. Suppose'the conditions of Lemma 4.2 are satisfied. Then for n.e.

gEG there is an ffF, I f 1 - ~ ' 2 ~ , such that b(g , f ) sp&,2" .

P r o o f . The assertion immediately follows from Lemma 4.2 and Lemma 4.1 setting A={(g , f ) : b(g, f ) z & 2 n } . Q.E.D.

Thus, the following construction of Saposhenko (cf. [2], [lo]) is possible for a.e. spanning subgraph gCG: Let f be such that b(g, f ) 5(p&$ and I f \ - ~ ' 2 ~ . Con- struct first an irredundant covering C, of B(g, f ) by maximal subcubes of g. The subcubes of C, cover at most (p&d2n+d1+0(2n) vertices provided that p Zn-'(') (cf. Theorem 3.5, where we have shown that for these probabilities subcubes of dimensions greater than dl=rlog log n -log log ( l / p ) ] cover only o(2") vertices). The remaining vertices we can cover by maximal embedded d-cubes. Using these d-cubes an irredundant covering C, of f - B(g, f ) covers at most I f I vertices. There remain at least

% s 2 n - 1 f I - q&d2n+d1 -o( 2")

non covered vertices gEE"-(f UB(g, f ) ) which can be covered by maximal embedded d-cubes in at least

(%d (1-0(1))y


different ways. It is clear that, possible omitting certain members of C, and C,, all these different ways yield different irredundant coverings of g . Thus, we have established the following

Lemma 4.4. 8uppose the conditions (4.2) are satisfied, and let p tn-"('). Put d , = flog log n -log log ( l / p ) l . If

and (4.3) &d2d' -0

(4.4) P"0 I

(4.5) irredundant coverings containing at least (4.6) 2" (1 - o ( l ) ) maximal subcubes.

Evidently, (4.6) yields the asymptotic Z+(g)-2" for a.e. gCG since Z+(g) s 2 " is trivially fulfiled. In the following we will see that the Iower bound (4.5) gives the logarithmica,l asymptotic for the number t(g) of irredundant coverings of a.e. spanning subgraph g.

Lemma 4.5. Let p =p(n ) such that condition (2.5) is satisfied. Suppose d =d, +3, where d2 i s defined as above. Then

then a.e. spanning subgraph gEQ has at least

( fjjd (1 -o( q))-"(l))

Proof . Pixe a vertex GEE''' and recall that D?wd=Ewi-fjjiij2d. Let us bound d -'

Erca= 2 a, Prob, , where a, = (a) (t) (:If) is the number of ordered pairs s=o

(I<! R') of d-cubes of En containing the arbitraryly fixed vertex a and having exactly an s-cube in common. Prob, denotes the probability that such a fixed pair consists of maximal embedded subcubes of h = (9, f). For s 2 1 we find Prob, s

, where this upper bound is the probability that such a fixed pair (R, K ) consists of embedded (not necessary maximal) subcubes of h=(g, f ) . Further Prob,S(zz')z (1 - 2 y + y ' / ~ ) " - - ~ ~ : Prob (KU UK'&g)=z2, Prob ( K U K ' S f U a ) = z ' z and Prob (R and K are maximal) 5

Ipd2d-s28- i 'Zd+1-2s-1 - --s,s-' - P - (zz')2 p P p-2a i-

2 z ( l - - p + p ( 1 - - p ( d i - 2 ) Z d - l - t ) 2 ) n-d . Th us, a o P r o b o c c ) ( z ~ ' ) 2 ( 1 - 2 y f y 2 / p ) ~ - ' ~ S

=ii$ ( i + o ( l / p n ) ) since for d=d2+3 we have y=p(df2)2d-1- -

and using that 2 a,sd (a,+ad) d

5 lz;

s=i =o(l/n). Setting as=

Q.E.D.

314 Nath. Nachr. 120 (1985)

Theorem 4.6. Let p = p ( n ) satisfy (2.8) and suppose d=d2+3 . Then for a.e. g E G there are (4.8) t ( g ) = (@;-'(1))?

irredundant coverings C of g of length 1(C) - 2".

Corollary 4.7. Suppose the conditions of Theorem 4.6 are satisfied. Then for a.e. gEG! we have (4.9) log t(g) - 2"d log n .

d2d- Z Proof of t h e Corol la ry . Recallthat ad= z , wherez=p for d = d 2 + 3

lies between llnc and lln. Since furthermore (2.8) yields d,-loglogn we have @ d - - - T L ~ ( ' - ' ( ~ ) ) . Now (4.9) follows from (4.8). Q.E.D.

Proof of T h e o r e m 4.6. Upper bound: For the number of all subcubes of a.e. gEG we derived the asymptotic i(g)-id,+2+id,+3 (cf. Theorem 2.6 (i)), where

2n-dz=@d2n--d. By studying the ratio ?jd+i /? jd it is easily seen that this

(2

asymptotic implies

i ( g ) ~ n % , 2 ~

(from now on cl stands for d, + 3). Since the subsets of subcubes corresponding to irredundant coverings are pairwise incomparable, for a.e. g

Lower bound: Recall that for d=d,+3 (l-z~)'+~-+l and thus iZa-Gdz'.

Regarding that - log log n s d s log log n we can choose p' =p ' (n) tending to 0

such that p '2dsn- ' (d) : for example p' with log (l/p') =o(log log n) is a suitable one. Considering that d, s l o g log n +log log log n, pp'n tn'-@') and Gd zn'(l-'(l))

condition (4.3) can easily be verified using the upper bound (4.7). Thus, by Lemma 4.4

1 2

t ( g ) Z ( G d (1 - o ( ~ ) ) ) ~ ~ ( ~ - o ( ~ ) ) ~ ( ~ ~ - o ( ~ ) ) 2 ~ - ,

Q.E.D. The proof shows that Theorem 4.6 allows the following reformulation :

Corollary 4.8. Suppose the conditions of Theorem 4.6 are satisfied. Then there exists a probability p' =p ' (n) -0 such that for a.e. g EG

log t ( g ) - 2" log Ed .

References

[I] P. ERDOS, J. SPENCER, Evolution of the n-cube. Comp. & &laths. with Appls. 5 (1979) 33 - 39 [2] -4. A. CAIIOXEHKO, 0 Hali6OJIbmefi ,WIliH€! TyIIZfKOBO$i a. H. (p. y IIOWITH BCeX &JHKqMa. MaTeNI.

3 a . i i e ~ m 4 (1968) 6, 649-658


[3] A. A. CAIIOXEHKO, JIeKqmi no reopmi M H H H M H ~ ~ ~ H X . Moscov 1972 (unpublished) [4] -, 0 CJIOHEIOCTH ~3 'bIOHKTHBHhlX HOpMaJIbHbIX (POPM, nOJIy4aeMbIX C IIOMOII(bl0 FpaAEIeHT-

[3] -, GH3'bIOHKTIIBHbIe HOPMaJlbHhle (POpMbI. XOSCOV 1975 Horo anropnma. C6. , , g y c x p e ~ ~ ~ & mama'' 21 (1972) 62-71

[6] K. WEBER, Random graphs-a survey Rostock. Math. Kolloq. 21 (1982) 83-98 [7] -, The length of random Boolean functions EIK 18 (1982) 12, 659-668 [8] -, Subcubes of random Boolean functions EIK 19 (1983) 7/8, 365-374 [9] -, Prime implicants of random Boolean functions EIX 19 (1983) 9, 449458

[lo] -, Irredundant DNF of random Boolean functions EIK 19 (1983) 10/11, 529-534

Wilhelm- Pieck- Universitat Sektion Mathematik

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Subcube Coverings of Random Spanning Subgraphs of the n-Cube

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