Random Subgraphs of Cayley Graphs overp-Groups

Europ. J. Combinatorics (2000) 21, 1057–1066 Article No. 10.1006/eujc.2000.0421 Available online at http://www.idealibrary.com on

Random Subgraphs of Cayley Graphs over p-Groups C. M. R EIDYS The subject of this paper is the size of the largest component in random subgraphs of Cayley graphs, X n , taken over a class of p-groups, Gn . Gn consists of p-groups, G n , with the following properties: (i) G n /8(G n ) ∼ = Fnp , where 8(G n ) is the Frattini subgroup and (ii) |G n | ≤ n Kn , where K is some positive constant. We consider Cayley graphs X n = 0(G n , Sn0 ), where Sn0 = Sn ∪ Sn−1 , and Sn is a minimal G n -generating set. By selecting G n -elements with the independent probability λn we induce random subgraphs of X n . Our main result is, that there exists a positive constant c > 0 such that for λn = c ln(|Sn0 |)/|Sn0 | the largest component of random induced subgraphs of X n contains almost all vertices. c 2000 Academic Press

1.

I NTRODUCTION

Random subgraphs of Boolean and generalized n-cubes play an important role in graph, coding theory and mathematical biology [6, 16]. In particular, connectivity and the largest component of random subgraphs of Qn2 [1, 7–9, 11, 13, 15] have been studied in great detail. The class of Cayley graphs considered in this paper is closely related to Boolean and generalized n-cubes. We will consider Cayley graphs X n = 0(G n , Sn0 ), where G n is a p-group and Sn0 ⊂ G n with the following properties: G n /8(G n ) ∼ = Fnp , where 8(G n ) is the Frattini K n 0 −1 group, |G n | ≤ n , K > 0, and Sn = Sn ∪ Sn , where Sn is a minimal generating set of G n . We will further assume that diam(X n ) ≤ n ` , for some ` ∈ N. For example, let G n = Fn2 , then 8(Fn2 ) = 1 and 0(G n , Sn0 ) ∼ = Qn2 , the Boolean n-cube. We have then Sn0 = Sn and n diam(Q2 ) = n. One main idea in the proofs is to retrieve information from the homomorphism G n → Fnp under which Sn maps into a basis of Fnp . In particular, this will allow for an analysis of the vertex boundary of X n -subgraphs of size ≤ n h for some h ∈ N. Selecting group elements g ∈ G n independently with probability λn we induce random subgraphs of X n . In this paper we will be interested in the largest component of random induced subgraphs 0n of X n . Our main result is, that there exists a constant c > 0 such that for λn = c ln(|Sn0 |)/|Sn0 | almost surely (a.s.) the largest component of random induced subgraphs of X n contains almost all vertices. 2.

P RELIMINARIES AND S TATEMENT OF THE M AIN R ESULT

Let X be a finite undirected graph with vertex set v[X ] and edge set e[X ]. Two adjacent vertices P, Q are called extremities of an edge y and a subgraph Y < X is called induced, if P, P 0 ∈ v[Y ] are adjacent in Y if and only if P, P 0 are adjacent in X . A path in X is a multiset (Q 1 , y1 , Q 2 , . . . , yn , Q n+1 ), where Q i ∈ v[X ], yi ∈ e[X ], Q i and Q i+1 are extremities of yi . The length of a path is the number of edges in the multi-set (Q 1 , y1 , Q 2 , . . . , yn , Q n+1 ); the distance d X (P, Q) between X -vertices P, Q is the minimal length of an X -path connecting P, Q or ∞ if there is no such path. The diameter of X , diam(X ), is the maximum of all distances in X . X is called connected if any two vertices occur in an X -path. X -subgraphs induced by maximal connected subsets of vertices are called components. Let Y be a subgraph of X ; the sets of X \ Y -vertices and X -vertices, that are adjacent to some vertices of Y , are called the Y -vertex boundary, d X Y , and Y -closure, Y , respectively. 0195–6698/00/081057 + 10

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A Cayley graph, 0(G, S 0 ), consists of a group G and a set S 0 ⊂ G that has the properties hS 0 i = G, S 0 −1 = S 0 and 1 6∈ S 0 . The vertex set of 0(G, S 0 ) is G and {P, Q} is a 0(G, S 0 )edge if and only if P Q −1 ∈ S 0 . The random graph model. Let (X n )n be a family of graphs. We denote the set of all induced X n -subgraphs by X (X n ) and for 0 ≤ λ ≤ 1 and 0n ∈ X (X n ) we consider the probability space (X (X n ), µn ), where µn {0n } = λ|0n | (1 − λ)|X n |−|0n | . (X (X n ), µn ) is called the random graph. Suppose now that Pn is a property in (X (X n ), µn ) such that limn→∞ µn {Pn } = 1 holds. Then we say ‘Pn holds almost surely (a.s.) in (X (X n ), µn )’. Let G be a finite group. The intersection of all maximal subgroups H < G is a characteristic subgroup of G, the so-called Frattini subgroup 8(G). For M ⊂ G, an element g0 ∈ G is called a non-generator if hM, g0 i = G if and only if hMi = G. The Frattini group 8(G) is generated by all non-generators of G; in particular, if S is a minimal generating set we have 8(G) ∩ S = ∅. For a p-group G we have G/8(G) ∼ = Fnp and for a minimal generating set, S, |S| |S| = n holds and there is a natural projection πn : G −→ F p , g 7→ g8(G), under which S is mapped into a basis of the vector space Fnp . According to a theorem of Burnside all minimal generating sets for a p-group have equal cardinality. We will consider minimal Cayley graphs over the following class of p-groups Gn = {G n | G n is a p-group, [G n : 8(G n )] = p n ∧ ∃ K > 0 : |G n | ≤ n Kn }.

(2.1)

For G n ∈ Gn let Sn be a minimal generating set; taking Sn0 = Sn ∪ Sn−1 we obtain the Cayley graph X n = 0(G n , Sn0 ). Note that for p = 2: |Sn0 | = n and for p > 2: |Sn0 | = 2n. G n is embedded in Aut(X n ) as the group of right-multiplications ρg = (x 7→ xg). Let Sk (P) = {Q ∈ G n | d X n (Q, P) = k};

(2.2)

then we immediately observe that |Sk (P)| ≥

(

|Sn |
k |Sn |
k

2k

for p > 2 for p = 2 .

Qk Qk x ji x ji Let P = i=1 s ji (P) and Q = i=1 s ji (Q) be two vertices where x ji = ±1, and for i 6= h: ji 6= jh . Then { ji (P) | i = 1, . . . , n} 6= { ji (Q) | i = 1, . . . , n} H⇒ P8(G n ) 6= Q8(G n ).

(2.3)

E XAMPLES . • Take G n = Fn2 , where F2 is the finite field over two elements and S i k : F2 → Fn2 ; x 7→ (0, . . . , x, . . . , 0) the embedding into the kth factor. Let Sn = nk=1 i k (F2 \ 0), then 0(Fn2 , Sn0 ) ∼ = Qn2 , the so-called Boolean hypercube. Qn • Take G n = i=1 Z/Z p k , let Sn be a minimal generating set and X n = 0(G n , Sn0 ).
k Then, for p > 2, we have diam(X n ) = n p 2−1 and |X n | = p kn .

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By c, c0 , c1 , . . . and K , K 1 , . . . we will denote positive constants. Let Z n be a random variable (r.v.) that is a sum of independent r.v.’s, having values in {0, 1} [3, 10]: then we have for  2 cη = min − ln(eη [1 + η]−[1+η] ), η2 µn {| Z n − E[Z n ] | > η E[Z n ]} ≤ 2e−cη E[Z n ] .

(2.4)

Further, for any non-negative, integer-valued r.v. Z n we have E[Z n ] ≥ µn {Z n > 0} and E[Z n ]/` ≥ µn {Z n > `} and, in particular, lim E[Z n ] = 0 H⇒ lim µn {Z n = 0} = 1

n→∞

n→∞

holds. In this paper we will prove the following theorem. M AIN T HEOREM . Let X n = 0(G n , Sn0 ) with diam(X n ) ≤ n ` and let (X (X n ), µn ) be (1) the random graph with λn = c ln(|Sn0 |)/|Sn0 |. For 0n ∈ (X (X n ), µn ) let Cn be its largest component. Then we can choose c > 0 such that lim µn {0n | |Cn(1) | ≥ [1 − ]|0n |} = 1,

n→∞

or equivalently

|0n | ∼ |Cn(1) | 3.

∀ > 0,

a.s.

V ERTEX B OUNDARIES

We begin this section by presenting a theorem on expanders [2, 4, 5]. Then we will improve this result by estimating the vertex boundary of induced subgraphs Yn < X n of size |Yn | ≤ n h . T HEOREM 1 ([2]). Let G be a group, S a generating set and 0(G, S) be a Cayley graph. For an induced subgraph X < 0(G, S) with |X | ≤ |0(G, S)|/2 1 |d0(G,S) X | ≥ |X | (3.1) 2 diam(0(G, S)) + 1 holds. P ROOF. Let G be a group acting transitively on a set M with |M| = n and let A ⊂ M. Then, [5, 12]: 1 X |A ∩ g A| = |A|2 /n. (3.2) |G| g∈G

Now let M = G and X < 0(G, S) be an induced subgraph such that |X | ≤ |G|/2. G acts transitively (and regularly) on 0(G, S) and |X ∩g X |+|g X \ X | = |X |. Hence (3.2) implies the existence of at least Q L one g ∈ G with the property |g X \ X | ≥ |X |/2. Since S generates G we can express g = k=1 sk with L ≤ diam(0(G, S)). We observe that for each ζ ∈ v[X ]\v[g X ] Q L 0 −1  Q L 0  there exists a minimal index 1 ≤ L 0 ≤ L such that k=1 sk ζ ∈ v[X ] and k=1 sk ζ 6∈ PL v[X ], whence |g X \ X | ≤ sk |sk X \ X |. From this we conclude that there exists one sk0 with the property |sk0 X \ X | ≥ |X |/(2 diam(0(G, S))) and the proof of the theorem is complete. 2 L EMMA 1. For arbitrary h ∈ N, L > 0, λ > 0 there exists an m ∈ N such that for all Yn ∈ X (X n ) with |Yn | ≤ n h |{P ∈ v[Yn ]| |{Q ∈ S1 (P)| |S1 (Q) ∩ v[Yn ]| ≥ m}| ≥ Ln}| < λ|Yn | holds.

(3.3)

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P ROOF. We will prove Lemma 1 by contradiction. For k ∈ N, let X k+1 (L , m) = {P ∈ X k (L , m) : |{Q ∈ S1 (P) : |S1 (Q) ∩ X k (L , m)| ≥ m}| ≥ Ln}. Claim 1: Let k ∈ N. For all L 0 > L > 0, λ0 > λ > 0 and m ∈ N there exists an m 0 ∈ N such that |X 1 (L 0 , m 0 )| ≥ λ0 |Yn | H⇒ |X k (L , m)| ≥ λ|Yn | holds. Claim 1 is proved by induction on k; k = 1 is obvious and the induction hypothesis reads ∀L 0 > L 0 > L, λ0 > λ0 > λ, m 0 > m, ∃m 0 ∈ N such that |X 1 (L 0 , m 0 )| ≥ λ0 |Yn | H⇒ |X k (L 0 , m 0 )| ≥ λ0 |Yn | holds. Suppose now that we cannot perform the inductive step k 7→ k + 1: ∃m ∈ N, ∀m 0 ∈ N, ∃Yn :

|X 1 (L 0 , m 0 )| ≥ λ0 |Yn | ∧ |X k+1 (L , m)| < λ|Yn |.

Then |X k (L 0 , m 0 ) \ X k+1 (L , m)| ≥ (λ0 − λ)|Yn | and for P ∈ X k (L 0 , m 0 ) \ X k+1 (L , m) we have: • |{Q ∈ S1 (P)| |{S1 (Q) ∩ X k−1 (L 0 , m 0 )}| ≥ m 0 }| ≥ L 0 n, • ≤ Ln vertices Q ∈ S1 (P) have the property |S1 (Q) ∩ X k (L , m)| ≥ m. Accordingly, there are ≥ (L 0 − L)n vertices Q ∈ S1 (P) with |S1 (Q) ∩ X k−1 (L 0 , m 0 )| ≥ m 0 and |S1 (Q) ∩ X k (L , m)| < m. We next consider the set Z n = {(P, Q, P 0 )} where: • P ∈ X k (L 0 , m 0 ) \ X k+1 (L , m), • Q ∈ S1 (P) where |S1 (Q) ∩ X k−1 (L 0 , m 0 )| ≥ m 0 and |S1 (Q) ∩ X k (L , m)| < m, • P 0 ∈ S1 (Q) ∩ (X k−1 (L 0 , m 0 ) \ X k (L , m)). We now proceed by estimating the cardinality of Z n . For fixed Q we have |S1 (Q)∩X k (L 0 , m 0 )| ≤ m, whence there are ≤ m vertices P that induce Z n -elements (P, Q, P 0 ). Accordingly, we obtain (a) ∀ P 0 ; |{P | ∃Q; (P, Q, P 0 ) ∈ Z n }| ≤ m2n. For fixed P we next compute the number of elements (P, Q, P 0 ) ∈ Z n . In view of the graph automorphism ρ P −1 = (x 7→ x P −1 ) we can w.l.o.g. assume that P = 1 holds. Let (1, Q, P 0 ) ∈ Z n , then we have Q = si±1 with si ∈ Sn and |S1 (Q) ∩ (X k−1 (L 0 , m 0 ) \ X k (L , m))| ≥ [m 0 − m]. Thus there are [m 0 − m] − 2 vertices P 0 ∈ S1 (Q) ∩ (X k−1 (L 0 , m 0 ) \ X k (L , m)) such that P 0 = sk±1 si±1 , k 6= i. For P 0 = sk±1 si±1 we have S1 (P 0 ) ∩ S1 (P) = {si±1 , sk±1 } and for ±1 ±1 ±1 ±1 6= s ±1 {i, k} 6= { j, h}, sk±1 si±1 8(G n ) 6= s ±1 j sh j sh 8(G n ) holds. Therefore we derive sk si from which we conclude that [m 0 − m] − 2 (b) ∀ P; |{P 0 | ∃Q; (P, Q, P 0 ) ∈ Z n }| ≥ (L 0 − L)n . 2 Combining (a) and (b) we finally obtain |Yn | ≥ |{P 0 | ∃P, Q : (P, Q, P 0 ) ∈ Z n }| [m 0 − m] − 2 1 ≥ [λ0 − λ] |Yn | (L 0 − L)n , 2 2mn

∀m 0 ∈ N,

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which is impossible and Claim 1 follows. In particular, we have for L 0 > L > 0 and arbitrary m ∈ N, X k (L , m) 6= ∅. Obviously, we can w.l.o.g. assume that 1 ∈ X k (L , m) holds. Next let,   k Y x ji ∗ Sk = Q ∈ G n | Q = s ji where x ji = ±1, ji 6= jh for i 6= h . (3.4) i=1

Claim 2: Let L 0 > L > 0 and λ0 > 0, then for all m ∈ N there exists an m 0 ∈ N such that   n 1 ∗ k−r |X (L 0 , m 0 )| ≥ λ0 |Yn | H⇒ |S2r ∩ X (L , m)| ≥ . r ∗ Claim 2 is proved by induction on r . The case r = 0 is clear. A vertex P ∈ X k−r (L , m) ∩ S2r has the form 2r Y xj P= s ji i , where x ji = ±1, ji 6= jh for i 6= h. i=1

We can only multiply with generators sk±1 , k ∈ { j1 , . . . , j2r }, finitely many times. Therefore we can assume w.l.o.g. that all Ln vertices Q ∈ S1 (P) are obtained by multiplying with some sk , k 6∈ { j1 , . . . , j2r }, i.e., ∗ |S1 (P) ∩ S2r +1 | ≥ Ln. ∗ k−r −1 (L , m)| ≥ For Q ∈ S1 (P) ∩ S2r +1 and m ∈ N we can choose m 0 such that |S1 (Q) ∩ X m. We accordingly conclude that

∀m ∗ ∈ N

∃ m0 ∈ N :

∗ ∗ |S1 (Q) ∩ X k−r −1 (L , m) ∩ S2r +2 | ≥ m .

∗ there are ≥ Consequently, for each P ∈ X k−r (L , m) ∩ S2r

Lnm ∗ 2

vertices

P ∈ S1 (Q) ∩ X Q ∈ S1 (P).
2r +2 ∗ ∗ for which there exists a Each P 0 ∈ S2r vertices P ∈ X k−r (L , m) ∩ S2r +2 has ≤ 2 ∗ 0 Q ∈ S1 (P) ∩ S2r +1 such that P ∈ S1 (Q). Using the induction hypothesis, we derive     Lm ∗ n 1 ∗ k−r −1 |S2r +2 ∩ X (L , m)| ≥ n
2 r 2r +2 2   n ≥ . r +1 Therefore, by choosing m = m(r ) sufficiently large, the assertion holds for finite r . Since |Yn | ≤ n h , h ∈ N this is impossible and there exists no L 0 > 0, λ0 > 0 such that for arbitrary m 0 ∈ N there is a Yn with |X 1 (L 0 , m 0 )| ≥ λ0 |Yn |, proving Lemma 1. 2 0

k−r −1

∗ (L , m) ∩ S2r +2 ,

T HEOREM 2. Let X n be the Cayley graph 0(G n , Sn0 ), and let Yn be an induced subgraph of X n with |Yn | ≤ n h , h ∈ N. Then ∃n 0 ∈ N, K h > 0; ∀ n ≥ n 0 ;

| d X n Yn | ≥ K h n |Yn |.

P ROOF. Let Sn = {s1 , s2 , . . . , sn } and ` ∈ N, we set ( ` ) Y Mn,` = ski | ski ∈ Sn , ∀i, j ∈ N` : i < j H⇒ ki < k j ∧ ski 6= sk j . i=1

(3.5)

(3.6)

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Taking the images of the above
elements g under the homomorphism πn : G n → G n /8(G n ) we observe that |Mn,` | = n` . Claim 1: ∀1 ≥  > 0 ∃K > 0 such that there are ≥ d[1 − ]ne G n -generators sk , with the property |sk Yn \ Yn | ≥ K |Yn |. Suppose that ∃ 1 ≥  > 0 such that ∀K > 0 there are ≥ dne generators sk with the property |sk Yn \ Yn | < K |Yn |, for 1 ≤ k ≤ dne. W.l.o.g. we can assume that n ∈ N. Now, |Yn ∩ gYn | + |g X \ Yn | = |Yn | and (3.2) imply that X |gYn \ Yn | = |G n ||Yn | − |Yn |2 . (3.7) g∈G n

By assumption (|Yn | ≤ n h ) we can choose ` ∈ N such that Q` that K ` < 1/2. For g = i=1 ski we have |gYn \ Yn | ≤

` X

n
`

≥ n2|Yn | and K > 0 such

|ski Yn \ Yn |.

i=1

Q` elements g of the form g = i=1 ski , (3.7) implies that   X n |G n ||Yn | − |Yn |2 ≤ ` K |Yn | + |gYn \ Yn |. `

Since there are exactly

n
`

g∈G n \Mn,`

We immediately derive the contradiction     n n 1 2 |Yn | − |Yn | ≤ |Yn | ` ` 2 and Claim 1 follows. We fix  > 0 with 1/4 >  > 0. According to Claim 1 there exists some K > 0 such that ≥ [1 − ]n generators have the property |sk Yn \ Yn | ≥ K |Yn |. We now choose L 0 , λ0 > 0 such that L 0 + 2λ0 < K (1 − ) and observe, according to Lemma 1, that ∃ m0 ∈ N :

|X 1 (L 0 , m 0 )| < λ0 |Yn |.

The X 1 (L 0 , m 0 )-vertices have ≤ 2nλ0 |Yn | neighbors. Further, there are ≤ L 0 n vertices Q ∈ S1 (P) where P ∈ v[Yn ] \ X 1 (L 0 , m 0 ) with the property |S1 (Q) ∩ v[Yn ]| ≥ m 0 . Therefore, |d X n Yn | ≥

1 [(1 − )n K |Yn | − 2nλ0 |Yn | − L 0 n|Yn |], m0

and the proof of the theorem is complete. 4.

2

P ROOF OF THE M AIN T HEOREM

P ROOF. Claim 1: Suppose that h ∈ N, then for c2 > 0 there exist a.s. no 0n -components Cn of size c2 n ≤ |Cn | ≤ n h . Let U` be the r.v. counting the 0n -components of size `, ` ≤ n h . According to Theorem 2 every 0n -component of size ` ≤ n h has a X n -vertex boundary of ≥ K h n ` vertices and the

Random subgraphs of cayley graphs over p-groups

 event that this boundary is empty has probability 1 −

1063

 c ln(|Sn0 |) K h n` . |Sn0 |

X n is a |Sn0 |-regular

graph, whence there are ≤ (` − 1)! |X n ||Sn0 |`−1 different components of size ` in X n . Therefore,   c ln(|Sn0 |) K h n` ` 0 `−1 1− E[U` ] ≤ ` |X n ||Sn | |Sn0 | ≤ eln(`)` |X n ||Sn0 |`−1 e−c ln(|Sn |)K h ` 2 . 0

1

By assumption we have |X n | ≤ n Kn for some K > 0 and we compute h

n X

`=dc2 ne

h

E[U` ] ≤

n X

eh ln(n)` |X n |eln(|Sn |)` e−c ln(|Sn |)K h ` 2 0

1

0

`=dc2 ne 0

0

0

1

≤ n h eh ln(|Sn |)c2 n n Kn eln(|Sn |)c2 n e−c ln(|Sn |)K h c2 n 2 . Hence, by choosing c > 0 sufficiently large Claim 1 follows from h

lim

n→∞

n X

E[U` ] = 0.

`=dc2 ne

Claim 2: At least [1−n −cc6 ]|X n |-X n -vertices are in the vertex boundary of a 0n -component of size ≥ c12 n. Let P ∈ G n , |Sn0 | = d2λ−1 n e + n 2 and let Q ∈ 0n be one of the n 2 X n -neighbors of P. W.l.o.g. we can assume that P = 1 and consider a branching process initialized at Q = sk±1 : Qh • by hypothesis, Q h = i=1 sk±1 has been constructed. We select among the d2λ−1 n e i indices an index j 6∈ {k1Q , . . . , kh } with probability λn and, in case we have selected h ±1 ±1 s ±1 otherwise the process stops. i=1 ski j , we set Q h+1 = s j

All vertices Q h+1 , h = 1, . . . constructed in the above process are pairwise different since their images w.r.t. the homomorphism πn : G n → G n /8(G n ) are different. According to [14] these branching processes produce a 0n -component of size ≥ c P n with constant probability p > 0.

Let Tn be the r.v. counting the X n -vertices having less than 12 c ln(|Sn0 |) 0n -neighbors. We then have, (2.4) [3, 10],   1 0 ∃c1 > 0 : µn P has ≤ [c ln(|Sn0 |)] 0n -neighbors ≤ 2e−c1 c ln(|Sn |) . 2 By linearity of expectation there exists a c10 > 0 such that E[Tn ] ≤ n −cc10 |X n | and E[Tn ]/` ≥ µn {Tn > `} implies for c11 < c10 :

lim µn {Tn > n −cc11 |X n |} = 0.

n→∞

Suppose now that P is a vertex taken from the remaining [1−n −cc11 ]|X n |-X n -vertices (having ≥ 21 c ln(|Sn0 |) 0n -neighbors). The probability that none of its 0n -neighbors is contained in a 1 0 0n -component of size ≥ c P n is (1 − p) 2 c ln(|Sn |) = n −c3 c . Let Z n be the number of these vertices, then E[Z n ] = n −c3 c |X n |. Again, E[Z n ]/` ≥ µn {Z n > `} implies ∃ n 0 ∈ N : ∀n ≥ n 0 :

µn {Z n > n −cc5 n |X n |} ≤ n −c(c3 −c5 ) .

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Accordingly, we have for c5 < c3 limn→∞ µn {Z n > n −cc5 |X n |} = 0. We now choose c6 > 0 such that n −cc11 + n −cc5 ≤ n −cc6 and have therefore at most n −cc6 |X n |-X n vertices that either have ≤ 12 c ln(n) 0n -neighbors or are not neighbors to a 0n -component ≥ c12 n, whence Claim 2. According to Claim 1 there are a.s. no 0n -components of size c2 n ≤ ` ≤ n h , whence all other X n -vertices are either 0n -vertices or contained in the vertex boundary of a 0n component of size ≥ n h . Analogously we obtain lim µn {0n | ≤ λn n −cc6 |X n | 0n -vertices are not in a component of size ≥ n h } = 1.

n→∞

Taking 0n0 to be the induced subgraph of 0n induced by all vertices that are contained in a 0n -component of size ≥ n h , there exist constants c, c6 , such that: • every vertex of 0n0 is contained in a connected 0n -subgraph of size ≥ n h , • |0n0 | ≥ [1 − n −cc6 ] |X n |, • |0n0 | ≥ [1 − n −cc6 ] |0n |. The vertex set of 0n0 decomposes into a set B of at most d|0n |/n h e 0n -connected subgraphs W1 , . . . , Wm , where | Wi | ≥ n h . In a second random process we then select the vertices of c0 ln(|S 0 |) X n \0n0 with the independent probability λ0n = |S 0 | n . Suppose that {B1 , B2 } is a bipartition n of B with the properties min{d X n (P1 , P2 )|P1 ∈ B1 , P2 ∈ B2 } ≥ 2 ∧ c2 |X n | ≤ |B1 | < c0 |X n |, 0 < c2 ≤ c0 ≤ 1/2. Claim 3: ∃ c, c0 such that B1 and B2 are a.s. connected by vertices selected in the second random process. 0

h

Obviously, there are at most 2|0n |/n different ways to construct B1 . By assumption we have diam(X n ) ≤ n ` for some ` and according to Theorem 1 we have |d X n B1 | ≥

1 |B1 |. (2n ` + 1)

According to Claim 2, |X n \ 0n0 | ≤ n −cc6 |X n | holds and we can choose c sufficiently large such that c00 |(d X n B1 ) ∩ 0n0 | ≥ ` |X n |. n Since we have 0n0 = B1 ∪ B2 the (d X n B1 ) ∩ 0n0 -vertices connect B1 and B2 . Now, an edge of the form {P, Q} where Q ∈ B1 and P ∈ d X n B1 is selected with probability λ0n 2 and each P ∈ d X n B1 has at least 1 and at most |Sn0 | neighbors contained in B1 . From this we conclude 0 c00

that none of the above edges {P, Q} is chosen with probability ≤ [1 − λ0n 2 ] n`+1 derive for c0 sufficiently large and h = 4 + `:    0  c00  0 c2 |X n |   c ln(|Sn0 |) |X n |  0 |) 2  n `+1 c ln(|S 3 0 n lim 2 |Sn |n4+` 1− = 0. n→∞   |Sn0 |

|B1 |

. Now we

That is, the probability of keeping the bipartition {B1 , B2 } separated in the second random process tends to zero, proving Claim 3.

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Claim 3 immediately implies that the closure of the largest component, Cn has the property (1) ∃c > 0 : ∀ > 0 : lim µn {0n ||Cn | ≥ [1 − ] |X n |} = 1. n→∞

(1)

Let now Yn be the 0n -subgraph induced by all vertices of 0n \ Cn . Claim 4: We can choose c sufficiently large such that ∀ > 0 :

lim µn {0n | |Cn(1) | ≥ [1 − ]|0n |} = 1.

n→∞

Suppose that |Yn | ≥ c1 |0n |. Since we have |0n0 | ≥ [1 − n −cc6 ]|0n | the subgraph Yn0 = Yn ∩ 0n0 contains c10 |0n | 0n0 -vertices. For constant, large, c12 > 0 let L n be the r.v. counting the X n vertices that have > c12 c ln(n) 0n -neighbors. There are (using the large deviation results of sums of independent Boolean r.v.’s [3, 10] and E[L n ]/` ≥ µn {L n > `} for non-negative, integer-valued r.v.’s) ≤ n −c11 |X n | of those vertices. Note that c11 depends only on c12 . By choosing c12 sufficiently large we conclude from this that there are c20 |0n | Yn0 -vertices having degree at most c12 c ln(n). Let ∂ X n Yn0 be the set of edges {P, Q} that have exactly one extremity in v[Yn0 ]. Then |∂ X n Yn0 | ≥ c20 |0n |[|Sn0 | − c12 c ln(n)], which immediately implies that |d X n Yn0 | ≥ c13 |0n |. Now we remove from d X n Yn0 all X n vertices that have > c12 c ln(n) 0n -neighbors (being connected to at most |Sn0 |1−c11 |X n | edges) and obtain |d X n Yn0 | ≥ [c12 c ln(n)]−1 [c20 |0n |[|Sn0 | − c12 c ln(n)] − |Sn0 |1−c11 |X n |]. Thus |Yn | > c13 |X n |, which is a.s. impossible according to Claim 3, and Claim 4 follows completing the proof of the theorem. 2 ACKNOWLEDGEMENTS I gratefully acknowledge the proofreading and help of W Y C Chen and Q H Hou. Further, I want to thank C L Barrett for many discussions and one referee for pointing out some relevant references. Special thanks to Darrell Morgeson for his continuous support. R EFERENCES 1. M. Ajtai, J. Koml´os and E. Szemer´edi, Largest random component of a k-cube, Combinatorica, 2 (1982), 1–7.

2. D. Aldous and P. Diaconis, Strong uniform times and finite random walks, Adv. Appl. Math., 2 (1987), 69–97.

3. N. Alon, J. Spencer and P. Erd˝os, The Probabilistice Method, Discrete Math and Optimization, Wiley Interscience, Wiley, New York, 1991.

4. L. Babai, Local expansion of vertex transitive graphs and random generation in finite groups, in: Proceedings of the 23 ACM Symposium on Theory of Computing (ACM New York), Vol. 1, 1991, pp. 164–174. 5. L. Babai and V. T. Sos, Sidon sets in groups and induced subgraphs of Cayley graphs, Europ. J. Combinatorics, 1 (1985), 1–11. 6. B. Bollob´as, Random Graphs, Academic Press, New York, 1985.

1066

C. M. Reidys

7. B. Bollob´as, Y. Kohayakawa and T. Luczak, The evolution of random subgraphs of the cube, Random Struct. Algorithms, 3 (1992), 55–90.

8. B. Bollob´as, Y. Kohayakawa and T. Luczak, Connectivity properties of random subgraphs of the cube, Random Struct. Algorithms, 6 (1995), 221–230.

9. Y. D. Burtin, On the probability of the connectedness of a random subgraph of the n-cube (in Russian), Probl. Pereda. Inf., 13 (1977), 90–95.

10. H. Chernoff, A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Stat., 23 (1952), 493–509.

11. M. E. Dyer, A. M. Frieze and L. R. Foulds, On the strength of connectivity of random subgraphs of the n-cube, Ann. Discrete Math., 33 (1987), 17–40. P. Erd˝os and A. R´enyi, Probabilistic methods in group theory, J. Anal. Math., (14) (1965), 127–138. P. Erd˝os and J. Spencer, Evolution of the n-cube, Comput. Math. Appl., (5) (1979), 33–39. T. E. Harris, The Theory of Branching Processes, Springer, Berlin, 1963. A. V. Kostochka, A. Sapozhenko and K. Weber, On random cubical graphs, In Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity, Ann. Discrete Math., 51 (1992), 155–160. 16. C. M. Reidys, Random induced subgraphs of generalized n-cubes, Adv. Appl. Math., 19 (1997), 360–377.

12. 13. 14. 15.

Received 25 May 1997 and accepted in revised form 15 April 2000 C. M. R EIDYS Los Alamos National Laboratory, TSA-2, Mail-stop: M997, Los Alamos, New Mexico 87545, U.S.A. E-mail: [email protected]