On the diameter and radius of randon subgraphs of the cube

On the Diameter and Radius of Random Subgraphs of the Cube

B. Bollobis Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 76 Mill Lane, Cambridge, CB2 7SB, United Kingdom

Y. Kohayakawa Instituto de Matematica e Estatistica, Universidade de Sa"o Paulo, Caixa Postal 20570, 01452-990 Sa"o Paulo, SP, Brazil

T. Luczak* Mathematical Institute of the Polish Academy of Sciences, Poznah, Poland

ABSTRACT

The n-dimensional cube Q" is the graph whose vertices are the subsets of (1,. . , n}, with two vertices adjacent if and only if their symmetric difference is a singleton.-Clearly Q" has diameter and radius n. Write M = n2"-' = e(Q") for the size of Q". Let Q = (Q,): be a random Q"-process. Thus Q, is a spanning subgraph of Q" of size t, and Q, is obtained from Q,-, by the random addition of an edge of Q" not in Q,_ , . Let t (*) = ~ ( 0 ; 6 2 k) be the hitting time of the property of having migimal degree at least k. We show that the diameter d, = diam(Q,) of Q, in almost every Q behaves as follows: d, starts infinite and is first finite at time t"), it equals n + 1 for t ( ' ) 5 t < t (* ) , and d, = n for t 2 6'). We also show that the radius of Q, is first finite for t = t"), when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Q, for t = O(M). 0 1994 John Wiley & Sons, Inc.

* On leave from the Department of Discrete Mathematics, Adam Mickiewicz University, ul. Matejki 48/49, Poznari, Poland.

Random Structures and Algorithms, Vol. 5, No. 5 (1994) 0 1994 John Wiley & Sons, Inc. CCC 1042-9832/94/050627-22

627

628 BOLLOBAS, KOHAYAKAWA, AND CUCZAK

1. INTRODUCTION

Let Q" be the n-dimensional cube, the graph whose vertices are the subsets of [n] = (1, . . . , n} and where two such vertices are adjacent if and only if their symmetric difference is a singleton. Note that both the diameter and the radius of Q" are n. Write N = 2" = lQ" l for the order of Q" and M = n2"-l = e(Q") for the size of Q". Let 0 = (Q,)," be a random Q"-process. This is a Markov chain whose states are spanning subraphs of Q" and Q, (1 5 t 5 M ) is obtained from Q,-, by the addition of an edge of Q" not in Q,, with this edge chosen uniformly at random from all the possibilities. We are interested in the behavior of e for large n; thus we use the terms "almost surely" and "almost every" to mean "with probability tending to 1 as n+w."

If P is a nontrivial monotone increasing property of spanning sujgraphs of Q", we let T~ = T ( P ) = T( e; P) be the hitting time of P in the process Q = (Q,),", that is,

* T~ = T ( P ) = T( Q ; P ) = min{t: Q, has P } .

The events P we shall consider here are, among others, (i) the event (6 2 k } that the minimal degree should be at least k , (ii) the event {diam < w } that the graph should be connected, (iii) the event {rad f r } that the radius should be at most r , and (iv) the event {diam f d } that the diameter should be at most d .

One of our main results implies that almost surely -

T( 0; 6 2 1) = T( Q; connectedness) = T( 0; diam 5 n + 1) = T ( a; rad = n) (1)

and

~ ( g ; diam = n) = T ( Q ; S 2 2 ) .

The fact that the hitting time of connectedness and ~ ( 6 2 1) almost surely coincide was proved as a remark in [3], where the hitting time of the existence of a perfect matching was shown to equal ~ ( 6 2 1) almost surely. Other results concerning matchings in random subgraphs of the n-cube are due to Kostochka [12], who investigated in great detail the size of the maximum matching in binomial random subgraphs of Q". (See also Kostochka [13] for generalizations.)

It follows from the relations above that the diameter of Q, in a typical 0 = (Q,)," behaves in the following very simple way. At the beginning of the process the diameter is infinite, and, when the isolated vertices disappear, it becomes finite for the first time, when it assumes value n + 1. Then it decreases to n = diam(Q"), its minimal possible value, exactly when the vertices of degree 1 disappear. As to the radius of Q,, we see that it becomes finite exactly when the isolated vertices disappear, when it assumes value n = rad( Q").

The results above are obtained as simple corollaries to results concerning the almost surely unique largest component, the "giant" component (see [17, 18, 191, and for more sophisticated results see [l] and [5]), of Q, for t = R(M). These results, namely, Theorem 10, Corollary 11, and Theorem 13, describe in detail the behavior of diam(l,) and rad(L,), where L, = L,(Q,) is the giant component of Q,, as t = R(M) grows.

Roughly speaking, Corollary l l ( i ) states that for all fixed k , in a typical

DIAMETER AND RADIUS OF RANDOM SUBCRAPHS OF THE CUBE 629

process, diam (L,) changes from n + k to n + k - 1 at a sharply defined time, namely, t, = M(1 - 2-'',(1 - (logn)/n)). The behavior of diam (L,) at around these critical values t, is also obtained. [See Corollary ll(ii).] Note that the results above completely describe diam(L,) for any t = f l ( M ) . Theorem 13 shows that, perhaps a little surprisingly, the radius rad(L,) of L, is rather stable. Indeed, that result says that almost surely rad(L,) = n - 1 for all a ( M ) = t < t") =

Thus, typically, diam(L,) decreases steadily as t increases while rad(L,) stays constant for a long while at n - 1, until it increases to n at time 6'). This is in sharp contrast with the case of ordinary random graph processes =(GI) , . Indeed, as proved by Burtin (see [8,9]), if t = (o(1og n ) / n > ( ; ) and o = w(n)+ a as n + m, then almost surely diam (G,) - 1 I rad(G,) 5 diam(G,). Thus, in the later stages of an ordinary random graph process, the diameter as well as the radius of the evolving graph decrease gradually. Questions concerning the diameter of ordinary random graphs G, E %(n, p ) and G, E %(n, t ) have been extensively studied, and very precise results are now known for a wide range of p and t. The interested reader is referred to [2, Chap. XI and to Burtin [8,9].

Finally, we remark that the radius and diameter of random subgraphs of Q" were also studied in Mahrhold and Weber [16], Kostochka, Sapozhenko, and Weber [15], and Sapozhenko and Weber [18]. However, the model studied by these authors is the one known as the "mixed" model, where the subgraph of Q" is chosen by the random deletion of vertices and edges from Q". The best results for this model given in the articles above are the ones in [15], where the authors determine the radius of the giant component L up to an additive constant of 2, and the diameter of L up to an additive constant of 8. Let.us mention that this model is also studied in Dyer, Frieze, and Foulds [lo], where connectivity problems are investigated.

This note is organised as follows. In the next section we give the fundamental lemma, Lemma 2, which is the key to many of our results. In Section 3 we give three preliminary results, which are fully developed in the following two sections into our main theorems concerning the diameter and the radius of the giant component of Q,. In the last section we mention a problem closely related to the ones we address here.

.( 0; 6 2 1).

2. THE FUNDAMENTAL LEMMA

For the study of random Q"-processes, it is often convenient to look first at the binomial model Q , of a random subgraph of the cube Q". As usual, given a graph H and 0 5 p 5 1, we write %(H, p ) for the space of random spanning subgraphs H, of H such that every edge of H belongs to H, independently with probability p . For a set X and k 2 0 we let denote the set of all k-subsets of X .

Our fundamental lemma, Lemma 2, will be proved in a slightly more general form than needed in this article. This is done because this version requires only a little more work, and will be used in [7] to study connectivity properties of random subgraphs of Q" (Here we use Lemma 2 only with 1 = 0 . ) Before we proceed, recall that Q" has vertex set P([n]) = 2'"'. To state our result, we shall


denote by Q"[-" a generic graph obtained from Q" by the deletion of some vertices in such a way that both 0 and [n] are left in Q"[-" and, for every k ( 1 I k 5 n - l ) , no more than 1 vertices from [n]'k' are missing.

Let us start with the following simple observation concerning Q""".

Lemma 1. joining 0 and [n] .

The graph Q'"" contains at least n!(l - O(l/n)) paths of length n

Proof. Let us call a path P = u0uI . . . us in Q" proper if u,, = 0 and us E [n]'s'. Let ui denote the number of proper paths of length i in Q""". Note that, for 1 5 i < n , we have u, 2 ui-l(n - i + 1 ) - li!. Therefore,

where as usual (a)b = a(a - 1) + * (a - b + 1 ) . Hence

as required. 0

The cornerstone of our article is Lemma 2 below, which states that the probability that two large sets of vertices should not be connected by a short path in Qb-'] E (e(Q'"-'l, p ) is superexponentially small. The proof we give for this result is based on a simple "splitting" argument for the (large) probability p and the second moment method. Independently Fill and Pemantle [ l l ] have used the second moment method to prove results of this nature in Q"; in particular, they determined the oriented first-passage time between two vertices furthest apart in Q" using an "enhanced" second moment method. (For improvements of some results in [ l l ] , see [4]). Lemmas in the spirit of Lemma 2 below may also be found in Sapozhenko [17], Sapozhenko and Weber [18], and Toman [19]. With Lemma 2 in hand, the proof of the main results will follow a rather natural course, although we shall encounter several technical difficulties on the way.

In the proof below and in the sequel, we only assume our inequalities to be valid for large enough n.

Lemma 2. Let LEN be fixed, and suppose that O < E = ~ ( n ) l l and that (1oglogn)ln < p = p ( n ) < 1. Then, for aZl S C [n]") and T C [n](rrp l ) with IS/, / T I 2 n('")'2 , the probability that, in Qb-" E (e(Q'"'], p ) there is no S-Tpath of length n - 2 is bounded from above by exp{ -Epn(log n)/log log n } .

Proof. Let 9 = {P: P an S-T path in Q"[-" of length n - 2}, and let s = IS1 and t = \TI. For each u' E [n]'" and u " E [n]"-" such that u' C u", the graph induced by all vertices w E Q""" such that u' C w C u" can be identified with Q(n-2)[-r1. Hence, as s 5 n and st 2 n'+', from Lemma 1, we get

u = 1 9 1 ~ ( s t - s ) ( n - 2 ) ! ( 1 - 0 ( L / n ) ) r f s t ( n - 2 ) ! r n ! / 3 n 1 - ' .

Let w = w(n) with log log n 5 w = o(n) be fixed. Set p,, = wln. We shall first study


the space %(Q"[-'], p o ) , and, to emphasize this fact, we shall write Epo for the expectation and Ppo for the probability in this space.

Let X = X(Q&']) be the number of paths in 9 that are present in Q&". Then

and hence p is rather large. We shall now compute the variance g2 = c.'(X) = Var(X) of X and show that u2 = O ( p 2 / w n ' ) . Let us first note that the second factorial moment of X is

E ~ ( X ) = Epo(X(X - 1)) = C C Pp,<f'~ Q C Q k ; ' ] ) P E P PZ4EB

= C Ppo(P, Q C Q;;']) + 2 Ppo<P, Q C Q L ; ' ] ) , 1 2

where C, indicates sum taken over all ordered pairs (P, Q ) E 9 X 9 of edge- disjoint paths P and Q, that is, such that E ( P ) n E(Q) = 0, and C, indicates sum over all the other pairs (P, Q ) . Clearly

C P,,(P, Q c Q:;" = C pp0(p c Q & ' ] ) P ~ , ( Q c Q;;']) - .= ( Ep0(X),' 1 1

Let us now estimate the sum over ordered pairs of edge-intersecting paths (P, Q ) . For k 2 1 and P E 9, let PP, , = { Q E 9: (E(P) f l E(Q)( = k } , and set

In order to estimate S,, we shall give bounds on NP,, = (CPP,:l for fixed P E 9. For any Q E PP,, there is an integer vector i = i(Q) = (i,), with 1 5 i, < - - * <

i , 5 n - 2 such that the i,th edge of P and Q coincide for all 1 zsj I k . Moreover, given i = (2,): as above, there are at most

k - 1

N~ = i l ! ( n - i, - I)! n <i j+ , - i, - I ) !

paths Q E 9 such that the i,th edge of P and Q coincide.

j = 1

Let i = (i,): as above be given. Define a = a(i) = (a,); by setting

if j = O ~ ~ + ~ - i ~ - l if l s j s k - 1

{ ; - i , - l i f j = k .

Note that then a,, a, L 1 , ai L 0 ( 1 5 j 5 k - 1 ) and C; aj = n - k . Moreover, k

Ni = n a,! = (n - k ) ! (a,: r.: 5 (n - k ) ! . j = O

Thus we have that

632 BOLLOBAS. KOHAYAKAWA, AND CUCZAK

and this crude bound will suffice for k 2 n/10. Let us now assume that k < n/10. We shall estimate NP,k in this range of k more carefully.

Let us write NP,k = N(ptL + N g i , where N(pt1 is the number of paths Q E Ppk such that if i = i(Q) and a = a(i) = (aj),", then aj 5 n - 3k for all 0 5 j k , and naturally N;; = N ~ , ~ - N ~ L .

Now, if maxO,j,k aj 5 n - 3k, then k k - 1 n a j ! s (n - 3k)! ma,( n bj: c bj = 2k} (n - 3k)!(2k)! ,

j = O j = O j

and so n - 2

NEL 5 ( ) (n - 3k)!(2k)! . (4)

To estimate NgL, note that if Q E Pp,k is counted in NgL, then, for i = i(Q), the vector a = a(i) = (aj)," is such that there is a unique j O (0 5 j o s k ) with ajo 2 n - 3k + 1 , since C j aj = n - k and k < n/10. The number of such vectors a = (a,)," with Ck aj = n - k is (k + l ) ( 3 k l '). Thus

3k - 1 N F i s (k + 1)( ) (n - k - l ) ! .

Therefore, putting together (3) , (4) and ( 5 ) , we have that for all k 2 1

and, if k < n/10, then

n - 2 s k 5 U[ ( ) (n - 3k)!(2k)! + (k f 1)(3kk- ' ) ( n - k ) ! } pi(n-2)-k .

Hence, recalling that p = Ep,(X) = UP:-' and u 2n! /3n1- ' , we see that

and

- 2 3n1-' c ( I ) ( n - 3k)!(2k)!pik

+ 2 (k + l)23k-1(n - k - l ) ! p i k

c s k 5 p { 1 s k<n / 10 n! I i k < n / l O

l d c < n / l O

DIAMETER AND RADIUS OF RANDOM SUBGRAPHS OF THE CUBE 633

Thus c.'(X) /p2 = O( 1 lwn') and by Chebyshev's inequality we have Pp,(X = 0) =

We now turn to the original space %(Q"[-'], p ) where p =p(n) is as given. Write H for a fixed H = Q"'-']. Let us generate a random spanning subgraph G of H by letting G = Uf==, HE' , where K = L p n / w ] , w = w(n) = log logn, p o = w/n, and the H E ) E %(H, p o ) are chosen independently. Note that p 2 Kpo, and hence it suffices to prove the claimed upper bound for G. By the inequality above, the probability that there is no S-T path of length n - 2 in G is at most {0(1/ wn')} 5 exp{ -epn(log n)llog log n}, completing the proof of our lemma. 0

O(l/on').

Before we can apply Lemma 2, we need to deal with some technical details concerning the distribution of the neighbors of most vertices in Qp E % ( e n , p ) . We start by introducing some further notation and terminology. A subcube of Q" is a subgraph induced in Q" by a set of the form

Qs.A = Q(S, A ) = {xE Q": x f l S = A } ,

where A C S C [n]. In the sequel we shall denote the subcube induced by Q,,, by Q,,, = Q(S, A ) as well, since this will not cause any confusion. The dimension dime,,, of Qs,A is n - IS/. Given x, y E Q" define the subcube (x, y ) = Q((xQy)', x n y ) , where uc = [n]b and as usual Q denotes symmetric difference. Thus, (x, y ) is the set of all subsets of x U y that contain x n y . In particular, if x C y , then ( x , y ) equals the "interval" [x, y ] in Q" = 9([n]) regarded as a partial order under inclusion, that is

(x, y ) = [x, y ] = {z E Q " : x C z C y } .

In what follows, given a subcube Q = (x, y ) , we may apply the automorphism u E Q" HXAU E Q" of Q" to translate Q so that it is mapped onto the interval [0, xQy]. In other words, it will suffice to consider subcubes Q = (x, y ) with x = 0. Now let G C Q" be a spanning subgraph of Q", and let u, u E Q" be given. Let Q = ( u , u ) and let G ' = G[Q] be the graph induced by the vertices of Q = ( u , u ) in G. As usual, denote the degree of a vertex z E G' by dG.(z). Then, we say that Q = ( u , u ) is (u , u)-good for G if dG,(u) , dG,(u) ?m2'3 , where m = dim Q is the dimension IuAul of Q.

In the sequel, if J is a graph and x, y E J , we write d,(x, y ) for the distance between x and y in J . In particular, if x, y E Q", then dQn(x, y ) = IxAyl =

dim(x, y ) . Moreover, as usual, if z E J , then T(z) = T,(z) is the set of neighbors of z in J , and, if 2 C V(J) , then T(Z) = T,(Z) = UrEZ T,(z). We may now state a simple technical lemma.

Lemma 3. Let x, y E Q" with dQ,(x, y ) 2 n/5 beJixed, and suppose N, C TQn(x),

634 BOLLOBAS, KOHAYAKAWA, AND WCZAK

N,,CT,,(y) are such that INXI, INyI 22n2I3. Let l / l o g l o g n c p = p ( n ) < l , and write Pxy for the following property concerning Q,, E %(en, p ) . (P,,) There is a vertex xo E N, and yo EN,, U rQp(Ny) such that dQn(xo, y o ) I d,n(x, Y ) + 1 and ( ~ 0 , Y O ) is ( ~ 0 , y o k o o d for Q p .

Then the probability that P,, holds is at least 1 - e~p{LR(n~'~/log log n)}.

Proof. We shall analyse two cases. For convenience, considering the automorphism u € Q" ~ x A u E Q" of Q", we may and shall assume that x = @. Since d,,(x, y ) 2 n/5 2 2n213, one of the following two cases must occur.

Case 1 . I [ x , y ] n N,I 2 n213: Choose C = Ln213] vertices x l r . . . , x, in [x , y ] n N,. Clearly there are C vertices y 1, . . . , y , E N,, such that x, C y, for all 1 I i 5 C. Note that d,,(x, , y , ) = Ix,Ay,I = ly,k,I 5 lyl = dQ,(x, y ) . Let us consider the subcubes Q, = [x , , y, ] . Then the probability of the event A , that Q, should not be ( x , , y,)-good for Q p E %(en, p ) is at most 2P(X 5 ~ 2 ; ' ~ ) = exp{ -R(n/log log n)}, where X is a binomial random variable with parameters p and rn, = dim Q,. The events A , (1 I i 5 4') are independent and so we have that the probability that PXy fails is exp{-R(nC/log log n)> = e x p { - ~ ( n ~ / ~ / l o g log n)}.

Case 2. IN,\[x, y]1> n213: Let N,\[x, y ] = { x l , . . . , x e } , where C = IN,\[x, y ] l 2 n213. We shall analyze two subcases. Suppose x, = x U {e ,} = {e , } (1 5 j 5 C). Now let N l = N,, n { Z E Q": Iz( = lyl + l}, and similarly NY = N,, n { z E Q": ( z ( =

Iyl-l}. Suppose N ; = { y : , . . .y ,+) , where y , + = y U { f ; } ( l s i s r ) . Let I = { e j : 1 I j I L'} n {i: 1 I i 5 r}. Now, if 111 2 n , then Case 2(a) below holds. On the other hand, if [ I [ < n213, then IN; U ( {h: 1 5 i 5 r}\{e,; 1 5 j 5 C})[ L n2I3, and Case 2( b) holds.

2 / 3

Case 2(a). There are C ' L ~ ' ' ~ vertices y,, . . . , y,, E N ; such that for all 1 si 51' there is a j = j ( i ) (1 l i s t ) such that xi C y , . Note that j = j ( i ) is uniquely defined for each i, that is, the map i- j = j ( i ) in injective. Let us consider the cubes Qi = [x j ( , ) , y i ] (1 I i 5 C'), and note that as in Case 1 above the probability that Px,, fails is exp{ -R(n5/3/log log n)}.

Case 2(b). There are C' L n213 vertices y , , . . . , y,, E N, such that x j g y i for any 1 I i 5 Cl and 1 i j 5 1. Let 1 5 i 5 C'. We shall say that the vertices y, U {e , } E T,,,(y,) (1 5 j 5 C ) are i-suitable. Note that the number of i-suitable vertices is clearly C. Let p o = p / 2 and pick Q,, E %(Q", p o ) randomly. The probability of the event B, that rQ ( y , ) should not contain at least n113 i-suitable vertices is at most

PO

where S,, ,,, is a binomial random variable with parameters C and p o . The events Bi are independent and hence clearly the probability that no B, occurs is e x p { - ~ ( n ~ / ~ / l o g l o g n ) } .

Suppose Bio occurs. Let y i , . . . , yk. be the io-suitable vertices adjacent to y i , in Q,,,). We know that C"2n113. For each 1 s k s C", let 1 I j = j ( k ) 5 C be the


unique j such that xjCy;. Note that then k + + j = j ( k ) is an injective map. Consider the subcubes Q k = y i ] for 15 k 5 i!?" and add edges to Qp, independently, each with probability p / 2 . It follows that the probability that Pxy should fail for Q p E %(Q", p ) is exp{ - f l @ ~ ~ / ~ / l o g log n)} .

Therefore in Case 2 we again have that the probability that the condition Pxy fails is as small as required.

3. PRELIMINARY RESULTS

In this section, we apply Lemmas 2 and 3 to show that if p is not too small, then a.e. Q , E % ( Q " , p ) is such that any two of its vertices of large degree are connected by a path of length at most n. We also show that this holds for a.e. Q, E % ( e n , t ) , where as usual %( Q", t ) is the space of all spanning subgraphs of Q" with t edges, all such graphs being equiprobable.

Let x , y f Q " be fixed. For all O l d s n and O < p = p ( n ) < l , write % x , y , 4 ( Q " , p ) for the conditional probability space obtained from %(Q", p ) by conditioning on the event { d (x), d Q p ( y ) 2 d } . We now use Lemmas 2 and 3 to show the following result concerning %x,y,d( Q", p ) . We remark that an analogous result for random Boolean functions may be found in Sapozhenko [17].

Q P

Lemma 4. Suppose 3/log log n S p = p ( n ) < 1 and d = d(n) = 2n2I3. Let x, y E Q" be two fixed vertices in Q". Then, with probability 1 - exp{ -fl(n log log n ) } , in the space %x,y,d( Q", p ) we have

213 dQp(x' y ) = dQn(x, y ) if dQn(x , y ) - 7

dQp(x, y ) I dQ,(x, y ) + 4 if dQn(x, y ) 2 n/5 , (6) dQp(x, y ) 5 0.8n if d,,(x, y ) I n/5 .

Proof. We may and shall assume that x = 0. Let S C rQn(x), T C T,,(y) be fixed and suppose 0 < p < 1. Let %ss,T(Q", p ) be the space obtained from %(Q", p ) by conditioning on the event {re (x) = S , r Q p ( y ) = T } . To prove our lemma, it is enough to show that if IS(, ITIp?2n2/3, then, for Q p E %JQ", p ) we have that d g p ( x , y ) satisfies ( 6 ) with probability 1 - exp{ -O(n log log n)} . Let us then fix S and T as above, and proceed to prove this assertion.

Let Po = p , = p 2 = l/log log n. Without loss of generality x = 0. We analyze three cases.

Case 1 . d,,(x, y ) = ( y l ? n - Set S' = S rl [x, y ] , T ' = T fl [x, y ] . Then IS'(, (T'I 2 n 2 I 3 . Hence, since [x, y ] can be obviously identified with Q m for some m 2 n - 2n , we may apply Lemma 2 to the triple [x, y], S', T'. Thus, we conclude that in Q p , E %s,T(Q", p o ) there is an x-y path of length d,,(x, y ) with probability 1 - exp{ -fl(n log log n)} .

2 / 3

Case 2. n /5 ~ d , , ( x , y ) = IyI <n - n2I3: Let Nx = S, Ny = T , and pick Qp, E %s,T(Q", p o ) randomly. According to Lemma 3, we know that property Pxy holds


with probability 1 - exp{ -fl(n4'3/log log n)} . Hence we may and shall assume that Pxy does hold for Qp,. Let xo and y o be as in Lemma 3. We now pick Qp, E%s,,(Q", p I ) randomly, and apply Lemma 2 to (xo, y o ) , which is an m-dimensional cube for some m with d,,(x, y ) - 3 I m 5 d,,(x, y ) + 1. Thus we find that in Qp, U Qp, there is an x-y path of length at most d,,(x, y ) + 4 with probability 1 - exp{ -n(n log log n)} .

Case 3. d,,(x, y ) = lyl< n/5: Pick Qp, E %s,r(Qn, p o ) . Let 2 = { z C [n]\y: Iz( = [n/51}. Then d,,(x, z ) = [n/51, and n/5 5 d,,(y, 2) 5 n/2. Moreover, the random variables dapo(z) ( z E 2) are independent and are such that dQpO(z) 2

2n2'3 with probability 1 - o(1). Thus, with probability 1 - exp{ -c"} for some c > 1, there is a vertex z E Q" such that n/5 5 d,,(x, z ) , d,,(y, z ) I n / 2 , and dQpO(z) L 2n213. We now pick Qp, E Ys,,(Q", pi) (i = 1,2), and argue as in Case 2. 0

Lemma 4 has the following immediate corollary.

Corollary 5. Suppose 3/log log n 5 p = p ( n ) < 1. Let do be the event that all x, y E Q p E %(Q", p ) with dQp(x), d Q p ( y ) m2n2I3 are such that (6) in Lemma 4 holds. Then the probability that Qp E %(Q", p ) satisfies do is 1 - exp { - n(n log log n)} .

Proof. Q p for x and y . Then

For x, y E Q", let d ( x , y ) be the event that (6) from Lemma 4 holds in

p(d0 fails) I P ( ~ x , y E Q": d ( x , y ) fails, dQp(x) , d Q p ( y ) 2 2n213)

I 22" max P ( ~ ( x , y ) fails, dQ,(x) , d Q p ( y ) 2 2nZ'3

= 22fl max P ( ~ ( x , y ) fails I dQP(x) , d , ( y ) 2 2n213)

* , Y

* . Y

x P(dQp(x), dQp(Y> 2 2n213)

= exp{ -n(n log log n)} . 0

We now consider &"-processes 0 = (Q,),", and show that vertices of large degree in Q, are a.s. connected by a short path as long as t is not too small.

Corollary 6. Almost every random Q"-process 0 = (Q,): is such that, for every t = t(n) 2 4M/lo log n and every two vertices x, y E Q" such that both of them have at least 2n E l 3 . neighbors in Q,, we have

dQl(x, y ) = dQn(x , y ) if dQn(x, y ) 2 n - n213 , dQI(x, y ) I d,,(x, y ) + 4 if d,,(x, y ) B n/5 ,

d,,(x, y ) 5 0.8n if dQ,(x, y ) 5 n /5 . (7)

Proof. This result is a straightforward consequence of Corollary 5. Let 4M/ log log n I t = t(n) I M be fixed and p = p ( n ) = (1 - ((log n ) / M ) ' 1 2 ) t / M . Then


p = p M = (1 -((log n)/M)l12)t = (1 + o(l))t, and, from standard estimates for binomial random variables,

where e(Q,) denotes the number of edges in Q,. In particular,

t - 2(M log n)"2 I e( Q,) I t (8)

holds for every 4Mllog log n I t 5 M with probability 1 - exp{ -(log n)/4 log log n} . Let %J Q", p) be the conditional probability space obtained from % ( e n , p) conditioning on the event given in (8). Note that if d is any event concerning Q, E %(Q", p), Then Pc(d) 5 (1 + o(l))P(d), where P, denotes the probability in %=(en, p). Let do be the event that (6) is satisfied for all x , y E Q, with dQ,(x), d , (y) Then, by Corollary 5,

Pc(do) = 1 - [ 1 + o(l)J exp{ -R(n log log n)} = 1 - exp{ -R(n log log n)} .

Now, we may generate Q, E %(Q", t) by first picking Q, E %c(Q", p), and then randomly adding t' = t - e(Q,) new edges to Q,. One may check that Q,. E % ( e n , t') is such that its maximal degree A(Q,.) satisfies Ace,.) 5 n213 with probability 1 - exp{ -R(n4'3/log n)}. Thus Q, satisfies the property that (7) of our lemma holds for all x , y E Q, with dQt(x), d Q t ( y ) 2 3n213 with probability 1 - exp{ -R(n log log a ) ) . Thus such a property holds for all Q, in 0 = (Q,): with t 2 4M/log log n with probability 1 - M exp{ -R(n log log n)} = 1 - exp{ -n(n log log n)}.

4. THE DIAMETER OF A R A N D O M SUBGRAPH OF THE /I-CUBE

In this section we shall study the behavior of the diameter of the almost surely unique largest component L = L(Q,) of Q, E %(&", t) when t = R(M). In fact, our main results will be "global" ones, in the sense that they will concern random Q"-processes 0 = (Q,),", and they will describe the behavior of diam(L) as t = R(M) grows.

It follows from Corollary 6 that, to estimate the diameter of the largest component L of Q, from above, it is enough to show that each vertex in L of small degree is within a small distance from some vertex of large degree. Our argument below makes this statement precise. To formulate and prove our results, we need to introduce some further definitions. Lemmas 7, 8, and 9 that follow are technical results needed in the proof of one of the main results of this section, Theorem 10.

Let H be a spanning subgraph of Q", and let P, = x o x l . . . x e , , P2 = yoyl . . . y t2 be two paths in Q" such that yo = x i = [n]L, and 1 5 k = t', + tZ. We sometimes refer to the pair (P,, Pz) with P, , Pz as above as a k-pair. We say that ( P I , P z ) is a k-stretching pair in H if (i) PI and Pz are paths in H and, moreover, (ii) the only edges of H incident to any of x , , . . . , xt l are the ones of P, and, similarly, the only edges of H incident to any of y, , . . . , ye, are the ones of Pz.

BOLLOBAS, KOHAYAKAWA, AND CUCZAK 638

The idea here is as follows. Suppose (P , , P 2 ) above is a k-stretching pair. Clearly we have d,(x,, y o ) 2 dQn(xo, y o ) = n, and the two paths P,, P2 “stretch out” from the “core” of H , giving two vertices, namely, x,, and y 1 2 , that are far apart. Indeed, we certainly have d H ( x e l , ye , ) 2 n + f l + f2 = n + k. Thus the existence of k-stretching pairs is an obstruction for the property of having diameter less than n + k. Our main aim is to show that a.s. this is the only such obstruction for any fixed k.

We remark that the very basic idea that stretching pairs are related to the diameter and radius can also be found in Kostochka, Sapozhenko, and Weber [lS], although in [lS] the exact relationship as given in our Theorem 10 below is not established.

Before we proceed, we need a few more definitions. Let PI = xoxl . . . x,, , P , = y , y , . . . yt2 for a k-pair of paths of Q“. If PI and P, satisfy (ii) above, we say that ( P , , P,) is a potentially k-stretching pair in H . If (P,, P,) is potentially k-stretching but not k-stretching, that is (i) does not hold, then we say that it is strictly potentially k-stretching. Finally, if ( P I , P 2 ) above is k-stretching in H and furthermore dH(xo) , d H ( y o ) 2 n/log n, then we say that (P , , P 2 ) is a proper k-stretching pair.

For k = 1 , 2 , . . . , let Yf) (respectively Yp)) denote the property that a spanning subgraph of Q” has at least t 2 M ( l - 2-”(k+1)) edges and contains no k-stretching pair (respectively, no potentially k-stretching pair). Clearly Yp) is an increasing property, whereas 9’;) is not. Our next result shows that Yt) is, however, “almost surely increasing,’’ and furthermore a threshold function for 3’F) is also a threshold function for 9:). For k 2 1 and a Q”-process 0 = (Q,),”, let tf’ = t f ) ( 0) be the hitting time of property Yf), where a = s, p. Thus

tf) = ( a ) - l l ( k + l ) t , ( Q ) = min{t I M ( l - 2- ): Q, has 3’:)) .

Lemma 7. n 2 1. Then almost every random Q”-process 0 = (Q,): satisfies the following.

Let k 2 1 be fixed and let w = w(n) -+ 03 be such that w 5 log n for all

(i) We have

M(I - p ( 1 - 1 ‘og:“n-w))

(ii) For a = s and p, the graph Q, has property Y f ) whenever t L t f ’ . (iii) If M( 1 - 2-1‘(k+‘)) 5 t < t f ) then there is a proper k-stretching pair in Q,.

Proof. Suppose t = (1 - 2-”,(l - (logn + C ) / n ) ) M , where C = C(n) = o(v%), and set p = p ( n ) = t/M. Let X f ) = Xf’(Q,) and xf‘) = XP’(Q,) denote the number of ordered pairs P , , P2 of Q“-paths such that (PI, P 2 ) is k-stretching and, respectiveIy, potentially k-stretching in Q,. Let us estimate E(Xf’). To this end, let us first fix a k-pair ( P I , P 2 ) of paths of Q”. Note that for this pair to be a k-stretching pair in Q,, we need that certain k edges of Q” should be in Q,, and that certain k, = kn + 0 ( 1 ) edges of Q” should not be in Q,. (Here and in the


sequel, the constants implicit in the 0-, 0-, and @-notation are allowed to depend on k.) Thus, the probability that (P, , P 2 ) should be a k-stretching pair in Q, is

= ( l + o ( l ) l ( l - ~ ) t k l t k (M) = ( l - p ) k ' p k = @ ( ( l - p ) k n ) .

The number of k-pairs of paths in Q" is clearly @(2"nk), and so we get

E($) = @(2"nk(1 - plkfl) = @(ePkc) , (10) where for the last equality we use that C = o(-. Similar calculations give that E(XP') = @(ePkc). Now the upper bound in (9) follows from (10) and Markov's inequality.

To see the lower bound for Tf) , let us suppose that C = C(n)+ --a, and as above C = o ( f i ) . Let us show that the number X = X(Q,) of k-stretching pairs ( P I , P z ) in Q, E Ce(Q", t) with P, a path of length k, and P2 a trivial path, is concentrated around its expectation. Let P = x , , x , . . . x k be a k-path in Q". We write E ' ( P ) for the edges of Q" that have at least one endpoint in {x,, . . . , x k } . Also, we write X, = X,(Q,) for the indicator function for the event that (P, , xi) should be a k-stretching pair in Q,. Then X = C, X,, where the sum extends over all paths P C Q" of length k. Note that

E2(X) = E(X(X - 1)) 5 (E(X))' + S ,

with S = C,, P(X,X, = l ) , where the sum is over all pairs (P, Q ) of distinct k-paths in Q" such that E ' (P )nE ' (Q)#0 . It can be easily checked that if XpXQ = 1 then at least k + 1 vertices in V ( P ) U V(Q) have degree at most 2. Thus

and hence

(11) n 2 k + l S = O(2 n )maxP(X,X, = 1 ) + 0 ,

where the maximum is naturally taken over all pairs (P, Q ) of distinct k-paths in Q" such that E'(P) n E ' ( Q ) # 0.

ThusVar(X) = o(E(X)~) since C = C(n)+ --a,, and so by Chebyshev's inequality we have that X = (1 + o( l ) )E(X) almost surely. Therefore, a.e. Q, E %(Q", t) is such that X = O(E-kC) when C = C(n)+ ---a, and C = C(n) = o ( f i ) . More- over, note that (11) above implies that a s . any two k-stretching pairs (P, x:) and (Q, y i ) in Q,, where P = xoxl . . . xk and Q = y ,y , . . . y k , are such that E'(P) f l E ' ( Q ) = 0 . To prove the lower bound for tf), let us consider a Q"-process 0 = (Q,): as a process where random edges of Q" are successively deleted from the evolving graph Q,. Let again t = (1 - 2-1/k(l - (logn + C)/n))M, where C = C(n)+ --a, and C = C(n) = o ( f i ) , and condition on X = X(Q,) = @(e-kc)+ w.

We claim that a.s. at least one k-stretching pair ( P , x i ) present in Q, will still be present in Q,,, where t' = rM(1- 2-1/(k+1) )1. Note that Q,, is obtained from Q, by the random deletion of a certain number of edges. Note also that the probability that no edge in E(P) is deleted in this process, where (P, xg) is a fixed

640 BOLLOBAS, KOHAYAKAWA, AND ~ U C Z A K

stretching pair in Q,, is bounded away from 0. Since X - m , our claim follows, and the lower bound for tf’ is proved.

(ii) Let us now consider property Yr) for Q,(t 2 tr’). Since Yf) is increasing, trivially Q, contains no potentially k-stretching pairs for t L tp’. As remarked earlier, property Yf’ is not increasing and hence we need to do a little work to deduce that Yt’ a s . holds for t 2 tr’. Set

)> 1 log n - log log n to = M(1 --p (1 - n

and note that E(Xf’) = E(Xp)( Q,,)) = @((log Q ) ~ ) . Also, as shown above, almost surely tf’ L to , and hence we may condition on this event.

Since by Markov’s inequality almost every 0 = (Q,),” is such that Xf’(Q,,) 5 (log log n)(logn)k, we may condition on 0 satisfying this property. We claim that almost surely no strictly potentially k-stretching pair (P, , P 2 ) of Q“-paths in Q,,, is k-stretching in Q, for t ? t O Clearly, this proves that a.s. Q, contains no k- stretching pair for t 2 t f ) , as required. To check the claim, fix a strictly potentially k-stretching pair ( P I , P 2 ) in Q,”. Suppose, as usual, P, = xoxl . . . x,, and P2 =

y , y , . . . y e , , where xo = y i = [n]\yo. Note that ( P 1 , P2) will become k-stretching in some Q, (t > to ) only if one of the edges of Q“ in E ( P , ) U E(P2) note present in Q,, is added to our evolving graph before any other edge incident to x,, . . . , x t l , y , , . . . , ye, . Now, clearly (E(P, ) U E(P,)( = k, and the total number of edges of Q“ incident to x,, . . . , x,,, y , , . . . , ye* is kn + O(1). Thus the probability that an edge of E((P,) U E(P,))\E(Q,,) should be added to our evolving graph before any other edge incident to the x, and y j is O(1ln). Hence the probability that a strictly potentially k-stretching pair in Q,, will ever turn into a k-stretching pair is at most O((1og log n)(log ~ ) ~ / n ) . This shows that Q, in 0 = (Q,): has no k-stretching pair if t I) f f ) almost surely, as required.

(iii) This follows from second moment calculations analogous to the ones in 0 (i). We omit the details.

The result above tells us that the critical values of t for the existence of stretching and potentially stretching pairs are t, = [M(1 - 2-1’(k+1) (1-(logn)/ n))) . We now look more closely at Q, when t is near such values.

Lemma 8.

Then

Let k 2 1 be fixed. Suppose t = M(1- 2-’Ik(l - (logn + C)/n) ) , where C = C(n)+casn+m. LetA =4.2- ’Ik( l -2- I l k ) k (1 +(k - 1)2-1-1’k)e-ck.

n+m lim P(tf’ 5 t ) = exp{ -A} . (12)

Proof. Let Xf’ = Xr’(Q,) count the number of k-stretching pairs in Q,. Then calculations similar to the ones in the proof of Lemma 7 concerning the random variable X = X ( Q , ) show that, for any fixed r L 1, we have limn+- E,(XI;“’) = A‘. Thus Xf’ converges in distribution to a Poisson random variable with mean A. (See, for instance, Theorem 20 in Chapter I of [2].) Thus (12) follows.

We now give a brief sketch of the calculations. Let p = t / M . Below we consider k-pairs (P, Q), where as usual P = x o . . . x t l , Q = y o . . . y t 2 . Let us write X(p,Q)


for the 0-1 indicator r.v. of the event that (P, Q) should be a k-stretching pair in Q,. To avoid double counting of the pair {P, Q}, in the k-pairs below we assume that, say, 1 E X , . It is easy to see that the total number of such k-pairs is (1 + 0(1))2"-'(k + l )nk . Among these, (1 + o(l))2"nk pairs (the ones with tl or

, and for the remaining t2 = 0) are such that E(X(p,Q)) = (1 + o( l))pk( 1 - p ) . Hence (1 + o( 1))2"(k - l)nk we have E(X(p,Q)) = (1 + o( l))pk( 1 - p )

Now we fix r l 2 . To estimate the rth factorial moment E,(Xf ' ) , we consider r-tuples U = ((P1, Q,) , . . . (Pr, Q,)) of k-pairs. In the sequel, we write C* to denote sum over all r-tuples U as above with all the r entries distinct, and C' for sum over all such r-tuples with at least one pair of repeated entries. Also, if P, = x i ' . . . x${,) and Q, = y t ) . . . y${,, form a k-pair (P , , Q,) (i E { 1,2}), then we say that (P17 Q,) and (P2 , Q 2 ) are distant if all the distances between a vertex in V ( P , ) U V ( Q , ) and a vertex in V(P,) U V ( Q 2 ) are at least loglogn. Below we write C, to denote sum over r-tuples U with the (Pg, Q,) (1 I i I r) pairwise distant, and C, to denote sum over r-tuples U with r distinct entries and with at least a pair of entries that are not distant. Then

n k - Z k + l

nk - 2k + 2

E ( x ~ ) ) = c ( ~ , ~ ) E ( x ( ~ , ~ ) ) = A + O(I) as n + w .

Moreover, clearly {E(Xf))}r = C, E ( X ( p l , Q r ) ) * * *E(X(p, ,Q,)) . To show that E,(Xf ' ) = A' + o( l), it suffices to show that

Now note that (i) and (ii) follow easily from the fact that C' and C, are sums over a "small" number of k-pairs U . To finish the proof, note that (iii) may be checked by considering the number of vertices of degree at most two in U, V(Pi) U U j V(Q,), as in the proof of (11). Indeed, for each r-tuple U , we consider a maximal subsequence of k-pairs ((Pi,, Q,,), . . . , (Pir,, Q j r , ) ) of pairwise distant entries (1 s i, < - * - < i,. 5 I), and note that if X ( p l : Q l ) - * * X(p, ,Q,) = 1 then there are at least r'k + 1 vertices of degree at most two in U i V(Pi) U U j V( Q j ) . However, the number of such U is at most 2nr'n0(10g log ") , and hence (iii) follows [cf. (11)J. 0

Before we can prove the main result of this section, we need to introduce one final piece of terminology. Suppose T,, T2 C Q" are two subtrees of Q", and x,, y o E Q" are such that dQ,(T, , x,) = dQ,(T2, yo) = 1. Thus T,, but x, is adjacent in Q" to some vertex of T , , and similarly for y o and T2. Suppose further that IT,( + IT2[ 2 k + 1, and dQn(x,, y o ) 2 n - k for some integer k 2 1. Let us say in this case that (T, , T2; x,, y o ) is a (k + 1)-system in Q". Let H C Q" be a

642 BOLLOBAS, KOHAYAKAWA, A N D CUCZAK

spanning subgraph of Q". Then we say that the ( k + 1)-system (T , , T,; xo, y o ) is a weakly ( k + 1)-stretching system in H if all vertices x E V(T,) U V(T,) have degree dH(x) < 3n213 in H. We then have the following simple lemma.

Lemma 9. Let k 3 1 be fixed. Suppose p = 1 - 2- l f k and t = LpMl . Then a.e. Q p E Ce(Q", p ) and a.e. Q , E %(en, t ) contains no weakly ( k + 1)-stretching system.

Proof. weakly ( k + 1)-stretching system in Q p is clearly at most

Fix a ( k + 1)-system ( T , , T,; xo, y o ) in Q". The probability that this is a

P, = {P(S"-~,~ < 3n2'3))k+' ,

where Sn-k ,p is a binomial random variable with parameters n - k and p . Indeed, we need at least k + 1 vertices of V(T,) U V(T,) to have degree less than 3n2I3, and for any vertex z in V(T,) U V(T,) there are at least n - k edges of Q" incident to z that are not incident to any other vertex of V(T,) UV(T,). Now, if C = L ~ I Z * ' ~ ) , then

P ( S n - k , p < 3 r ~ " ~ ) ~ 2 ( n - k ) (1 - P ) " - ~ - (

113 3n2" - (n-3nz i3)1k < n2n2'3.--n/k ~ 4 ( e n ) 2 -

Thus, since the number of (k + 1)-systems in Q" is at most 0(2nn2(k+1)) , we have that Q p E %( Q", p) contains a weakly (k + 1)-stretching system with probability

2-"" + l i k ) ) = 4 1 ) 3 0(2nn2(k + l))po = 0(2f l~z(k+l) ( l+n* '~)

as required. The statement for Q, E Ce(Q", t ) follows by the monotonicity of the property in question. 0

Let us say that a spanning subgraph H C Q n of Q" satisfies property Bk if H has e ( H ) 2 M(1- 2-11(k+1) ) edges, it has a unique component L = L(H) of largest order, and, moreover, L has diameter diam(l) 5 n + k - 1 . For a random Q"-process 0 = (Q,),", let

t f ) = t y ) ( 0 ) = min{t 2 M(l - 2-11(k+1)): Q, has GBk} .

We can now state and prove the main result of this section.

Theorem 10. tr) = T f ' , and Q, has property gk for all t 2 t p ) .

Proof. First recall that a.e. Q"-process 0 = (Q,): is such that Q, has a unique largest component, the "giant" component, if, say, t r 2 M l n (cf. [1,5]). More- over, it is easy to check that, quite crudely, almost every 0 is such that if t 2 M/log log log n, then Q, has, apart from the largest component L, only components of order O(1og log n). Let us condition on our Q"-process having the properties above.

Now, if M(1- 2-1'(k+1)) 5 t = t(n) < t f ) , and P, = x l x l . . . xe, , P2 =

Let k 2 1 be fixed. For a.e. random Q"-process 0 = (Q,)," we have

DIAMETER A N D RADIUS OF RANDOM SUBGRAPHS OF THE CUBE 643

y o y , . . . ye, form a proper k-stretching pair, then both xo and y o belong to the

almost surely. Let us now turn to the reverse inequality. Suppose t 2 t t ) = $)( 0). We may

assume that t f ) 2 (1 - 2-'Ik ) M , and hence that Q, does not contain weakly (k + 1)-stretching systems either. According to Corollary 6, we may and shall assume that (7) of that lemma holds for all x, y E Q, with d,,(x), d,,(y) 2 2n213. Let x, y E Q, belong to the giant component of Q,. We claim that d , , ( ~ , y ) 5 n +

giant component of Q,, and, moreover, dQ,(xpl, yt2) 2 n + k. Thus t f ) 2 t k (s)

k - 1 .

Case 1. d,,(x), D,,(y) ?3n2I3. From the assumption that (7) holds for Q,, in this case we have d,,(x, y ) I n.

For the next two cases, let us consider the set W of vertices of Q, that have degree less than 3n2 /3 , and let H = Q,[W] be the graph induced by W in Q,.

Case 2. dQ,(x) < 3n213, and d,,(y) 2 3n213. Let T C H = QJW] be a spanning tree of the component C, of x in H. There is a vertex x o E L such that d Q , ( X O , T ) = I , and consequently such that dQ,(xo) 2 3n213. Thus dQ,(x, Y ) 5

d,,(x, xo) + d,,(x,, y ) 5 [TI + n. So we may assume that IT1 2 k, since otherwise we are done. However, as Q, does not contain weakly (k + 1)-stretching systems, we must have I TI = k, and, moreover, T = C, and T must be a path with x as one endpoint and the other endpoint must be adjacent to no in Q,. We conclude that Q, contains an induced path P = xoxl . . . x k , where xk = x. We may also assume that dQ,(xo, y ) = n , and hence that dQn(xo, y ) = n. Since there are no k-stretching pairs in Q,, there is a vertex z E Q", z #xo, . . . , xk, adjacent in Q, to some x i (1 5 i 5 k). But then dQ,(z) 2 3n213, and d,,(z, y ) 5 n - 1. Thus dQi(x7 y ) 5 n + k - 1, as required.

Case 3. d,,(x), d,,(y) < 3n2l3. Let C, and respectively C, be the components of x and y in H = Q,[W]. Let T, C C, be a spanning tree of C, (i = 1,2). There are vertices xo, y o E Q" such that dQ,(xo), d,,(y,) 2 3n213, and dQ,(xo, T,) = d,,(yo, T,) = 1. Note that lT1l, IT2/ 5 k, and hence if dQ,(xo, y o ) 5 n - k - 1, we are done. So we assume that dQn(x0, y o ) 2 n - k . Since (T, , T,; xo, y o ) cannot be a weakly (k + 1)-stretching system in Q,, we have lTll + IT2[ 5 k, and, as d,,(x, y ) I ITl] + IT,/ + dQ,(xo, y o ) , we assume that lTll + IT2\ = k. In fact, we may assume that T , = C , , and T , together with xo forms an xo - x path P, , and similarly T, = C,, and T2 together with y forms a yo - y path P,. Also, we may assume dQ,(x0, y o ) = dQn(xo, y o ) = n. Since (P,, P,) is not a k-stretching pair in Q,, there is a vertex z E Q", z g P j 7 adjacent to some vertex in Ti in Q,, for i = 1, say. However dQ,(z) 2 3n213 and dQn(z , yo ) < n, and SO dQ,(x, y ) 5 d , , ( ~ , 2) + d,,(z, y ) I n + k - 1, as required.

This finishes the proof of our result.

We can now explicitly describe the behavior of the diameter of the largest component of Q, as t = n(M) increases.


Corollary 11. Q"-process 0 = ((2,): h such that if

Let k z 1 be fixed. ( i ) Zf w = w(n) -+ w as n 3 m, then a.e. random

n 1 1

M ( 1 - 21/" (1 - log ; + ")) 5 t 5 M( 1 - 2l/fk-l) (1 -

then there is a unique largest component L = L(Q,) in Qt and diam(L) = n + k - 1.

(ii) Suppose C = C(n)-+ c as n -+ w, and t = t(n) = M( 1 - 2-'/k( 1 - (log n + C)/n) ) . As in Lemma 8, let h =4.2- ' lk( l - 2-'/k)k(l + ( k - 1)2p1-'/k)e-"k. Then a s . Q, E % ( e n , t ) has a unique largest component L = L(Q,) and, conditional on this event,

and

Proof. Statement (i)

lim P(diam(L) = n + k - 1) = e-* n-m

lim P(diam(L) = n + k ) = 1 - e-* .

follows from Lemma 7 and Theorem 10. Statement (ii)

" e m

follows from Lemma 8 and Theorem 10. 0

An immediate consequence of Corollary 11 is the followipg. Let 0 = (Q,)," be a Q"-process, and let k 2 1 be an integer. Write t(k) = t(k)( Q) for the hitting time of the property (6 L k } , that is,

t ( k ) = t (k) ( Q ; - 6 2 k ) = min{t: S(Q,) 2 k } .

Corollary 12. In almost every random &"-process 0 = (Q,):, we have

(i) 6 ' ) = T ( Q ; s 2 I) = T ( 0 ; d i a m s n + I ) , (ii) t(') = T( Q ; S 2 2) = T( 0; diam 5 n ) .

Proof. Let us first recall that, as proved in [3], the graph Q, in 0 = (Q,): almost always becomes connected at time t = &'). Thus statements concerning the largest component of Q, from that time onwards are in fact statements about Q, itself.

Now, almost surely t ( ' ) 2 (1 - w/n)M/2 for any w = w(n)-+ a, and hence t(') 2 I 6') almost always. Therefore, by Theorem 10, almost surely (i) holds.

To see the statement concerning ~ ( a ; d i a m % n ) , note first that a spanning subgraph H C Q" of Q" contains a l-stretching pair if and only if it has a vertex of degree 1. Thus almost surely t r ' ( 0 ) = t(2': If Mllog log n I t < t (*) , then almost sure1 there is a vertex of degree 1 in Q,, and hence t f ' 2 t ( 2 ) , and, moreover, if t = t( ), then there cannot be a l-stretching pair in Q,, and hence ti"' L t"). Thus we have t ( ' ) 5 t ( 2 ) = ty', and hence again by Theorem 10 almost surely (ii) holds. 0

Y

5. THE RADIUS OF A RANDOM SUBCRAPH OF THE /I-CUBE

In this section we turn our attention to the radius_rad(l) of the giant component of Q, [t = f l ( M ) ] in a typical random Q"-process Q = (12,):. As mentioned in the


Introduction, unlike in ordinary random graph processes, the radius of the evolving graph in a typical Q"-process is rather stable, as the following result shows.

Theorem 13. Let O < E <1/2 be a constant. Then almost every random Q"- process 0 = (Q,)," is such that, for every t = t(n) with to = [ E M ] 5 t < t") , there is a unique largest component L = L(Q,) in Q, and rad(L) = n - 1.

Proof. We claim that almost surely, for each vertex x of the giant component L of Q,,, there exists a vertex y that also belongs to L, and such that d,,(x, y ) = n - 1. Indeed, notice first that if y has degree at least log n in Q,,, then y belongs to the giant component of Q,,, as the second largest component of Q,, has order O(1). To see the claim, it now suffices to note that, almost surely, for any z E Q", there is a vertex y E rQ&) with degree at least log FZ in Q,,. Thus almost surely, for every t 2 to, the radius rad(L,) of the largest component L, of Q, is at least n - 1.

Let us now show that almost surely rad(L,) 5 n - 1 for all t 2 to. Almost every random Q"-process 0 = (Q,)," is such that if t = t") - 1, then Q, has a unique isolated vertex zo = z, ,(Q). In what follows, we condition on 0 having this property. Let z E Q " be fixed, and let us further condition on the event that { z , ( o ) = z } . It suffices to show that, under these conditions, almost surely rad(L,) I n - 1 for t 2 to, and hence let us now verify this assertion.

Let %z(Q", t ) and respectively 9((Q", p ) be the spaces obtained from %(Q", t ) and % ( e n , p ) by conditioning on the event that z should be an isolated vertex. We now note that Q,, in 0 = (Q,)," is simply a random element from 9i2(Q", to ) , and therefore we turn to this latter space. However, let us first set p = p ( n ) = (1 - ((logn)/M)"')E, and consider %z(Q", p ) . Let do be as in Corollary 5. Then

P(Q, E %*(en, p ) fails do) s P ( Q , E % ( e n , p ) fails d O ) / P ( d Q p ( z ) = 0)

= exp{-R(n loglogn)}(l -p ) -"

= exp{ -Q(n log log n ) } ,

and hence a.e. Qp E %=(en, p ) satisfies do. It is also easily seen that the vertex x = zc = [n]b and all vertices y # z of Q" within Q"-distance nllog n from z have degree at least 31z''~ in Qp E gZ(Q", p ) .

As in the proof of Corollary 6, we may generate an element from %z(Q", to) by first generating an element Qp from %2(Q", p ) , and then adding a suitable number of random edges to Qp. Proceeding this way, and using the above fact about property do, we may show that a.e. Q,,E%z(Q",to) is such that (i) any two vertices x , y of degree at least 3n2'3 in Q,, are such that (7) of Corollary 6 holds, and, moreover, (ii) the vertex x = zc and all vertices y # z within Q"- distance nllog n from z have degree at least 3n2'3. Let k = ~ l /10g2( l l ( l -€))I. From Lemma 9 we know that a.e. Q,, E %=(en, to) is such that (iii) Q,, contains no weakly (k + 1)-stretching systems.

It is now enough to notice that if (i), (ii), and (iii) above hold for Q,,, then for any graph H with Q,, C H C Q" in which d&) = 0 we have that dH(x, y ) I n - 1

646 BOLLOBAS, KOHAYAKAWA, AND WCZAK

for x = zc and any y E H distinct from z , as long as x and y belong to the same component in H . 0

One can read out the following corollary from the above result.

Corollary 14. Almost every random Q"-process & = ( Q , ) : is such that I

t ( l ) = T ( Q ; S 2 1) = T( &; r a d s n ) . 0

6. CONCLUDING REMARKS

Let & = (Q,)," be a random Q"-process. It is known that if t is a little larger than to = M / n , almost surely there is a unique largest component in Q, and, moreover, this component is of much larger order than the other components (see [l] and [5]). This component is usually referred to as the "giant" component. One very interesting problem we have not dealt with in this note is that of the de- termination of the diameter of the giant component of Q, for t a little larger than to. This problem might be harder than the problems we have addressed here: It is likely that finer methods than the ones used in the proof of Lemma 2 might need to be developed. Quite possibly, these methods would be based on the Kruskal- Katona theorem, or more generally on isoperimetric inequalities for the cube, or the martingale method. In particular, it would be interesting to settle the following specific question.

Problem 15. In a typical random Q"-process 0 = (Q,): , is ever the diameter of a component of Q, superpolynomial? 0

This question concerns Q , for t close to to; there are numerous interesting problems about Q , for larger t , beyond the hitting time of connectedness. In a sequel [7] we shall study Q, in this range. In particular, we shall prove a result implying that for almost every cube process the hitting time of k-connectedness is equal to the hitting time of minimal degree at least k. We shall also study the "mixed model" for random subgraphs of Q". Here a random subgraph Q p , , P e of the cube is chosen by independently deleting vertices and edges from Q" with probabilities 1 - p , and 1 - p , , respectively. We shall prove some results that are analogous to the ones in this article, and thereby we shall answer some questions of Kostochka, Sapozhenko, and Weber [14]. (See also [15].)

ACKNOWLEDGMENTS

The first author was partially supported by the NSF under Grant DMS-8806097, the second was partially supported by BID/USP, Project 30.01, and the third was partially supported by KBN under Grant 2 1087 91 01. The authors would like to express their heartfelt thanks to Boris Pittel for carefully reading the manuscript and making a number of detailed and insightful comments. They are also


indebted to an anonymous referee who drew their attention to [17,18,19] and to parts of [16], which appeared after this article was submitted.

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[12] A. V. Kostochka, Maximum matchings and connected components of random subgraphs of the n-cube (in Russian), Methods Discrete Anal. Study Boolean Function Graphs, 48, 42-70 (1989).

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[17] A. A. Sapozhenko, Metric properties of almost all Boolean functions (in Russian), Diskretny Anal., 10, 91-119 (1967).

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Received May 19, 1992 Accepted January 11, 1994

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On the diameter and radius of randon subgraphs of the cube