Random Graphs, Second Edition (Cambridge Studies in Advanced Mathematics 73)

=

(I

-

)lk

> •., exp {2

> p(k; A)e-(ý-'+k)n

nk-I

'l22/n-

(1 (+Q))

(1.14)

As a trivial consequence of (1.13) and (1.14) we find that if p depends on n in such a way that pn -> 2 as n -+ oo, where 2 is a positive constant,

1.3 Normal Approximation

9

then the binomial distribution with parameters n and p tends to the d Poisson distribution with mean 2. In symbols: Sn,p--P;. The factorial moments of SP and P;, are calculated without the slightest difficulty: E,(S,,,p) =

Pr(n)r and E,(P;,) = Z.

1.3 Normal Approximation In probabilistic graph theory we often need good estimates for the probability in the tail of the binomial distribution. The best known example of such an estimate is the DeMoivre-Laplace limit theorem [see Feller (1966, pp. 174-195) or R6nyi (1970a, pp. 204-210)]. We shall give it here together with some bounds valid for all values in a given range. A random variable is said to be normally distributed with mean puand variance a2 if it has density function 1 This distribution is usually denoted by N(y, a). We shall reserve 0(t) and (D(x) for the density function and distribution function of N (0, 1), respectively:

0(t) = (t)

2

e-t /2

and D(x)

/2/dt.

=

If x > 0 and 1>_ 0, then (see Ex. 7) 21+1 (-)l3""(2m

mI

M-11

-- 1)

•aX}~ 1{-*x}

I

<

2-(1 <

)

"3 3 . -(2m-l

+E -Im

(1.15

X2m

m=1

In particular, as x --* 1 -1

we have )(x) x

1 lOe272/2" /22Xx

(1.15')

Our first aim is to give upper bounds for the terms b(k; n, p) of the binomial distribution. The restrictions in the theorem below are for the sake of convenience.

10

Probability Theoretic Preliminaries

Theorem 1.2 Suppose pn > 1 and 1 < hqn/3. Then if k > pn + h, we have 2hirph3--2pqn qn p2n + h + h h exp 1 b(k; n, p) <

Proof We may and shall assume that k find

b(l; n, p) <

=

pn + h. From inequality (1.5) we

(epn'Z)'

[n/{I27rl(n

l)}]1/2

-

for every 1,1 •_ 1 < n - 1. Hence 112 b(k; n,p)

(27rpqn)

< (pn)k+l/2 (n

k)n-k+1

2

qn-h--,1/2

pn pn+h+l/2 X(q

qn-h

(pn +(h

h)q~-1

h -p-

pn+h)

x x1 qn-h

From inequalities (1.9) and (1.10') we obtain that 1 2 (27rpqn) / b(k;n,p) < exp

-(pn+h+ 4,() + (qn

-

h + 1)

h

h

2p2n2)

+ 2q 2 n 2

}

-+2q3)

The required inequality follows by expanding and simplifying the expression on the right and taking into account the conditions pn > 1 and 1• h < qn/3. 11 Theorem 1.2 gives us the following upper bounds on the probability in the tail of the binomial distribution. Theorem 1.3 With the assumptions in Theorem 1.2 we have h2 { Ppqn 1/2 1 - exp-2pqn

P(S',p Ž pn + h) < (S2r)

Proof By relation (1.11), if m _ pn + h then b(m+ 1;n,p) < b(m; n, p)

+, h +- 1) q(pnhph+q<

h q

h3 + pn

.

1.3 Normal Approximation

11

Hence P(Snp

pn+h)=

E

bNm;n,p) <

_2 b([pn+h];n,p).

m!pn+h

It is easily checked that (1 - 2)-' < (pqn/h){1 + (h/pn)} < (pqn/h)eh/pn, so the result follows from Theorem 1.2. It is often convenient to use the following weak but concise form of this result. D Corollary 1.4 (i) If 1 < h < min{pqn/10,(pn) 2 / 3 /2}, x(pqn)112 we have

P(ISp - pn[> h) < (pqn)/

2

2 e-h /2pqn -

"h

=

1eX/2 x

(ii) Suppose 0 < p < 1 and (pn)-/ 2 < P([Sn,p - pnl > epn)

then with h

< 6. Then -,2pnl3q+Elq.

<

Proof (i) To deduce this from Theorem 1.2 we have to note simply that

h3

h

pqn +_2log pY~n2

1

(ii) Theorem 1.2 gives that for h P(Sn,p Ž-pn + h) <

2

pqn pn2

71

<- 2 o 2 =

epn we have

1/2

33p 3

+ pfn2 \pqn pE epqn - Epn/6)q exp

< 2 e-p/3 exp {-l 1I]

21

<e-e~ pn/3q 2

0.22579.

exP2 q {.1

--

66n(-6

6q E -2pn/3q+

/q

< •

Analogous calculations yield P(Sn,p •-- pn - h) < I

ne

<

2

so the result follows.

,

L-

Another upper bound for the probability in the tail is due to Chernoff (1952). See also Okamoto (1958). For 0 < p < l and q = l-p, write Hp(x) for the weighted entropy function: Hp(x) = x log(p/x)+(1--x) log{q/(1x)},0 < x < 1. Note that Hp(x) = Hq(I - x). Then we have P(Sn,p Ž- k) •_ exp{nHp(k/n)}

for k >_pn

12

Probability Theoretic Preliminaries

and, equivalently, P(Sn,p •- k) • exp{nHp(k/n)}

for k < pn.

To see these inequalities, simply note that for u > 0 and k > pn, by Markov's inequality (1.1) we have P(Sn,p Ž- k) • eukE(euSlP) = e-uk(q + peu)n. The right-hand side attains its minimum at eu = qk/p(n - k) and so

P(Sn,p Ž!k) <

p(1n -k))k (n'!k'

( n

(pf)k

"I)' = exp~nHp(k/n)},

as claimed. For a different approach to proving Chernoff's inequalities, see Exercise 16. In fact, the same inequalities hold for the sum X of n independent Bernoulli random variables X 1., .... Xn, with P(Xi = 1) = E(Xi) = pi and pi = pn. Indeed, by Jensen's inequality, for pn < k and u > 0 we have n

P(X > k) <_ eukE(eux) = e-uk J(qi + pieu) <- e-uk(q + peu)n. i=1

As before, this implies that P(X Ž! k) •_ exp{nHp(k/n)}. Janson (2001) pointed out that from this it is easy to deduce that if X.. Xn are independent Bernoulli random variables, X = In=1 X, and E(X) = l then for t Ž 0 we have P (X

> A+ t)

exp <_

2 ( A t/3) )

and P (X

_

-

t)

_ex p

-2

As our next result shows, the bound in Theorem 1.2 is essentially best possible, provided h = o{(pqn) 2/ 3}. Theorem 1.5 Suppose pn

1 and k = pn + h < n, where h > 0. Put

1 +12(n 1- k)' S=1k Then I p ex > 2-7-q b(k;n ~,p)> b ~ pp >

h2 (n, 2pqn -2q2n2

p 3 -3p3n3

2pn It f

-h3

)

1.3 Normal Approximation

13

Proof Stirling's formula (1.4) gives

b(k; n, p)>

(pfl)k (

n

[n/{27zk(n

)n-k

ký''g

-

Hence • qn\

(27rpqn) 1/2 b(k 1,n, p) > e-fl (Pmnk+112

e-fl (I+

h)-k-112 ( pn

k/ 1

_)

1

k-n-1/2

qn/

From inequality (1.8) it follows that )

h -fl-(pn +h+

1 (27rpqn) /2b(k; n,p) > exp

-

2 2p 2 n + 3p3n3

h +p h2 2q2n2+ + 2q-2

+ qn-+

+(nh+)qn

h3 ) 3q3n3)} 3•

The required inequality follows by expanding the expression on the right and noting that h < qn and pn > 1.

L

The estimates in Theorems 1.2 and 1.5 can be put together to give an approximation of the binomial distribution by the normal distribution. From the formulae so far it is clear that the most convenient measure of k - pn is in multiples of the standard deviation ax = (pqn)'/ 2 . Theorem 1.6 [DEMOIVRE-LAPLACE] Suppose 0 < p < 1 depends on n in such a way that pqn = p(1 - p)n -* ooasn -+ oo. (i)

Suppose h, < h2 , h 2 -h --+ oo as n --- oo and lhlI+ h2z = o{(pqn) 2/ 3}. Put hi = xi(pqn) 1/ 2 , i = 1, 2. Then P(pn + hi

(ii) If 0 < h = x(pqn)11

2

<_

S,

•

pn + h 2 ) -• D(x2 ) -

D(xi).

= o{(pqn)2 /3 }, then

P(Sn,p • pn + h)

1 - D(x).

'-

In particular,ifx -- oo, then

P(S,,p > pn + h) -.x

1

e

X2/2 .

Proof Let us assume for the moment that 1 < hi < h 2 . Then with

14

Probability Theoretic Preliminaries

a = (pqn)1 /2 from Theorem 1.2 we have

P(pn+hi h_ Sn,p <_pn + h 2 ) _ =

{1 +o(1)} 1

(

1

ht <

{1 +o(1)}

e h2/22

k(h/6)/a _ {1 +o(1)}[(D(h

2 /a)-(F{(hi-

1)/6I}]

h•
=

{I +o(1)j'D(X2 ) - b(xi)},

where in the sums we assume that h runs over the values making pn + h an integer. An analogous lower bound is obtained by Theorem 1.5, so (i) is proved. Now to lift the restriction 1 < hl, note simply that the omission of two terms from Zh, •_h
->

o,

d

the (Sn,p - pn)/ pqn -- N(O, 1). This is a special case of the Central Limit Theorem (see Theorem 1.23, below). Occasionally we are interested in larger deviations than that covered by Theorem 1.6, and then the following easy bounds are rather useful. We leave the proof as an exercise. Theorem 1.7 (i) Suppose 0 < p <_1,e pqn >_ 12 and 0 < e •_ 1/12. Then P(ISn,p

- pn1 > e pn) •_ (P2pn)-/2e- 2pf/3

(ii) If uq > 2 and pn > 1, then P(S,,p >_ upn) <

(e/u)upn,

and if v _ e and v2 pn >_log v, then

P

Sn,p

1_

pn

log v)

<e

t

'Pn.

The probability in the tail of the binomial distribution has attracted much attention; the best bounds to date (which are essentially the best possible) are due to Littlewood (1969), whose paper is highly recommended to those who think that the binomial distribution is too simple to deserve study.

1.4 Inequalities

15

1.4 Inequalities One of the oldest tools in combinatorics is the inclusion-exclusion principle. Many extensions of this principle are useful in several branches of mathematics, especially in combinatorics, number theory and probability theory. Some of these extensions are used to estimate the number of primes less than a given numbers by going through the natural numbers and sieving out the multiples of numbers already found to be primes. In this section we establish several sieve formulae and deduce some probability theoretic consequences of them. For the rich theory of the general inclusion-exclusion principle, called the theory of M6bius inversion, the reader is referred to Rota (1964), Aigner (1980) and Goulden and Jackson (1983). Suppose for every fixed r we are able to estimate the factorial moments Er(Xn) of a sequence of non-negative integer valued r.vs (X,). Then, using certain sieve formulae, we may be able to conclude that the X, tend in distribution to some known r.v. The aim of this section is to present the sieve formulae needed for this purpose. We start with the following beautiful and simple result of R~nyi (1958). Related results are due to Galambos (1966) and Galambos and R~nyi (1968); see also Comtet (1974, pp. 189-196). Theorem 1.8 Let fl,f 2 ...

fk

A 2,....A, and let bl,b 2...

bk be real constants. Suppose

be Boolean polynomials in n variables A 1 ,

k

ZbiP{f(B

1

, B 2, ... , B,)} > 0

(1.16)

i=1

whenever B 1, B 2 ... , B, are events in a probability space such that P(Bi) = 0 or 1. Then (1.16) holds.for every choice of events B 1,B 2, ...,B, in every probability space. Proof A Boolean polynomial can be expressed as a union of complete products, that is polynomials of the form gj(A 1,...,An)

(

A n iA

~

J

{l..n}. I_

Here Ai denotes the complement of Ai. Hence (1.16) can be rewritten as ZcjP{gj(BI .... Bj)} _ 0, J

(1.16')

16

Probability Theoretic Preliminaries

where the summation is over all 2' subsets J c only on J and the bi.

{1._ n} and cj depends

n}, choose B1 .... B, such that P(Bi) = I if Given a set Jo • {1. i c JO and P(Bi) = 0 if i ý Jo. Then P{gj(BI, ... ,Bn)} = 0 unless J = JO and P{gjo(B1 .... B,)} = 1. Thus with this choice of the Bi the left-hand side of (16') is cjo. Consequently cj > 0 for every set J, so (16') does hold for every choice of the events B 1 ,..., B,. D] An equivalent formulation of Theorem 1.8 is that every extreme point of the convex set

{(P{gj(A

: (Ai)' are events} c 1R2n

1

has only 0 and 1 coordinates, with precisely one 1. Corollary 1.9 Suppose we have equality in (1.16) whenever P(Bi) = 0 or 1. Then equality holds.for every choice of events B 1 ,...,Bn. Let A 1 ,A 2 ... J =N ={1 In}...

A, be arbitrary events in a probability space (Q, P). For put

n

Aj=

Aj

and A 0

= Q.

jEJ

Furthermore, set pi = P(Aj) and Sr

r =0,1 .I

Zpi,p

n.

IJ =r

The usual inclusion-exclusion formula is an immediate consequence of Corollary 1.9:

A)

P

=

(-1)r+lSr.

(1.17)

(-1)IJI P(Aj) = 0.

(1.17')

Equivalently,

P (

Ai ) i /l

Y J[_>!

1.4 Inequalities To see (1.17) note that if P(A 1 )

P(A,)

=

(A~) U

-

Given sets J ( L ( AnAj, i E M = L-J:

P

I 1

Z(_1)r+lS,

U (Ai n Ai)}

N

{1.

=

P(A 1 ) = 1 and P(A1 +1 )

=....

0, where I> 1, then S, =

(,)

.....

.....

and (- i)r~l

Z

17

(() =

0.

)r (,i

n}, let us apply (1.17') to the sets

1 Y (-1)1'I +P(AI

n A.)

=

Z(-1)1'

Tlp~ju

'#0O

1*0

Since the probability above is non-negative and at most p.j, we have 0 ! Y

(--1) KJIlpK ! pj.

(1.18)

JcKL

K 4J

In fact, as shown in R~nyi (1970a), the inequalities (1.18) together with p0 1 characterize the functions pi. Thus if p, e [0, 1] for every set I c N,pO = 1 and (1.18) holds for all J c L ( N, then one can find a probability space (n,P) and events A 1 ,A 2 , ... ,A, such that

P(ni,

Ai) =

p, for every I

c

N.

A considerable strengthening of the inclusion-exclusion formula is due to Ch. Jordan (1926, 1927, 1934) [see also Fr~chet (1940)]. This formula incorporates results of Poincar6 and of Bonferroni (1936). In order to state it in a concise form, we shall say that a sum n

S = Z(-1)k+lXk k=1

satisfies the alternating inequalities if (-1), {s + E(1)kxk} Ž 0 k=1

for every 1, 1 = 0, 1._

n. (Note that in this case we have Xk >_ 0.)

Theorem 1.10 Let A 1, . , An be events in a probability space, let So, S1 . . Sn be as above, and denote by Pk the probability that exactly k of the Ai occur. Then n

k= Pkk (1)r+k

()Sr

r=k

and the sum satisfies the alternating inequalities.

(1.19)

Probability Theoretic Preliminaries

18

Proof By Theorem 1.8 it suffices to prove the assertions in the case when for some 1,0 < I < n, of the sets Ai have probability 1 and the others have probability 0. Then Sr

(1)

=

and

Pk = 60k

If < k then all terms in

(1.19) are 0 and if 1 = k then (1.19) is of the form 1 = 1 -0 +0-0 Assume that 1 > k. Then k+m

-

r=k r~

Pk

=k+mM

k/ \r}

L r(k

k!

r!

k~m( (O _I)r~ (I

k!

r=krk

If k + m

...

= 0 and

- k r-k

klm(

_1

rk(

-)

(r-k)!

1 then the sum is O and if O _ m • l

-

k then it is

Hence the sum is positive if m is even and negative if m is odd. Thus (1.18) holds and the sum satisfies the alternating inequalities. El Corollary 1.11 . Let X be a r.v. with values in {0, 1. be the rth factorial moment of X. Then P(X = k)

=

(_l)k+r E,(X)

n} and let Er(X)

k!

and the sum satisfies the alternating inequalities. Proof Let Ai be the event {X > i},

=

1.....n, and define the numbers

So, S,..., S, as above. Then clearly Sr = Er(X)/r!

(1.20)

and {X = k} is the event that exactly k of the Ai occur. Hence the result follows from Theorem 1.10. El In the proof above the value of X is the number of Ai's that occur. In fact (1.20) holds whenever X is defined from an arbitrary set of Ai's in this way. Corollary 1.12 Let X be a non-negative integer valued r.v. whose rth factorial moment Er(X) = E{(X)r} is finite for r = 1,... R. Suppose s,

1.4 Inequalities

19

t • R,k + s is odd and k + t is even. Then

{

k! •_P(X = k)

(--)k+rEr(X)/(r -k)!

E <

E(-1)k+rEr(X)I(r-- k)!

k!.

r~=k}

k!

Proof Set X(") = min{X,n}. Then lim,,Er(X(n)) - Er(X) for r = 1,...,R, and for n > k we have P(X(") = k) = P(X = k). Therefore Corollary 1.11 implies our assertion. I Corollary 1.13 Let X be a non-negative integer valued r.v. with finite moments. If lim Er(X)rm /r! = 0

r---oo

for all m, then

{-Z(-1)k+rEr(X

P(X=k)=

- k)!

k!

and the sum satisfies the alternating inequalities. One has similar results for probabilities of the form P(X >_ k). The following is the analogue of Corollary 1.11. The straightforward proof is omitted. Corollary 1.14 With the assumptions of Corollary 1.11 n r Er(X) nj Er(X) P(X - k) (ukIr -1) r! = i) (k-1).(r--k)!r r=k

r=k

and the sum satisfies the alternating inequalities. The results above can be extended to several sequences of events and to several random variables. We shall only state the analogue of Corollary 1.11. Let us say that a finite sum S

X"Z ....,iý

satisfies the alternating inequalities if Cl

Cm

i .

a

xil...,'a

-

20

Probability Theoretic Preliminaries

is at least 0 if for each j either cj = bj or aj < cj < bj and cj - aj is even, and it is at most 0 if for each j either cj = bj or a, <• cj •_ bj and cj - aj is odd. Corollary 1.15 Let X1 ..... Xm, be random variables with values in {0, 1.

n}

and let

E(rl

r,.) = E{(Xi)ri ... (X.)r,,}.

.....

Then P(Xi

=k,....Xm

=

""

k.)

r1 =ki

HT(r..

-

El k,!

ki)! m

r,, =k,,

i=1

and the sum satisfies the alternating inequalities. Here r k = E', ki.

=

Ei' ri and

If the random variables take non-negative integer values then the inequalities still hold. In general, the inequalities appearing in the corollaries above are the best possible. However, if we know some relationships among the moments then we may be able to find better inequalities. The next result is an example of this. Theorem 1.16 Let X be a non-negative integer valued r.v. with moments ml = E(X) and m 2 = E(X 2 ). If mi • m2 < 2ml •< 2, then P(X = 0) < 1 - (3m,

-

M2)/2,

-

m2)/6.

and if 2m, < m2 < 3m, < 6, then P(X = 0) •< 1 - (5m,

Both inequalities are best possible. Proof Set pi = P(X

=

i),i = 0,1 .

Then

00

Z(3i - i2 )pi = 3m,

- M2,

so -PO -•pi •-(3m1 1

- "12) +r I1 E-(i2 1

-3i + 2)pi.

1.4 Inequalities

21

Since 2i(i - 2) < 3(i - 1)(i - 2)

=

3(i 2 - 3i + 2), i = 1, 2,...,

we have

3

12 -••i--2 - 33 3i +~ 2)p, _ 1

i2 - 2i)pi = 3(M2 - 2ml). 1

Hence

I - P0 > '(3m, - m2) + max {0,

'(m 2 -

2ml)},

which implies our inequalities. It shows also that the first inequality in the theorem holds unconditionally-which is no surprise since it is a particular case of Corollary 1.12. To see that the inequalities are best possible, in the first case define the distribution of X by p0 = 1 - (3m, - m2 )/2, p, = 2 ml -- M2, P2

=

(m2 -

ml)/2, 0

=

P3

=

P4 .....

and in the second case by

(3m, - m2)/2, P3 = (M 2 - 2m,)/3, 0 = P4 = P5 = .... The conditions on m, and m2 imply that these choices are possible. It is easily checked that the moments are as required. D Po = 1 - (Sm, - m2)/6, Pi = 0, P2 -

The first inequality in Theorem 1.16 is exactly the inequality implied by Corollary 1.12. Note also that both inequalities are better than the Chebyshev type inequalities (1.2') and (1.3). Our next aim is to present a rather unusual but widely applicable inequality, the Lovdsz Local Lemma. Given independent events AI, A 2 , ... ,Am with P(Ai) < 1 we know that P(n_1 jAi > 0) is not 0 since it is exactly ihmtP(Ai). Now, if the Ai are not independent but each Ai is independent of a system formed by 'most' of the Aj then a beautiful theorem of Erd6s and Lov~isz (1975) may enable us to deduce that P(n,.m=tAi) > 0. This result has become known as the Lovcsz Local Lemma. Extensions of the Lovfis Local Lemma were proved by Spencer (1975, 1977); the next two theorems, which can be read out of the original proof in Erd6s and LovS1sz (1975), are further slight extensions. Theorem 1.17 Let A 1 ,A 2 ... Am be events in a probability space, J(1), J(2), ... , J(m) subsets of {1,2,..., m} and 71,Y2,...,Ym positive reals less than I such that for each i, 1 <_i <_ m,

(i) A, is independent of the system {Aj (ii) P(Ai) •!ýyj-IjvJ(i)(1 --"j).

j V J(i) U {i}},

Probability Theoretic Preliminaries

22 Then

Ž

(izj)

vIj=1(1 --

j) and P

A1,

j=~

j

"

n/

Proof We prove the assertions by induction on m. The case m = 1 is trivial, and in the proof of the induction step it suffices to show the second assertion for that implies

P

Aj) =P

(A

Aj)

P

Aj

>(1--7)P

(i j=

j=1 j2

1-T)).

j=2j=1

Assume that J(1) = {2, ... , d + 1} so A 1 is independent of Ad+3, Am. Since by the induction hypothesis P(n'3t 2A 1 ) > 0, the conditional probability P(Al nm=2 Aj) is well defined and Ad+3 ...

S)n

=

P(A2n

... nAm)

P(Al n Ad+ 2 n... n Am)P(Ad+2 n... n Am) P(Ad+2 n... n A,)P(A2 n... n Am) P(Al lAd+2 n... n Am) P(A2 n ... n Ad+1lAd+ 2 n... n Am) As A 1 is independent of {Ad+2,.... ,Am}, the numerator is precisely A 1 . We can estimate the denominator as follows: P(A2 n-... n Ad+l lAd+2 n"... n Am)

P (A 2 IA 3n...n Am)P (A 3 IA4 ... d+1

P..(Ad+ IAd+2 n ... n Am)>H(1 - 7)=j=2

Am)

(1-7),

jCJ(1)

where we applied the induction hypothesis to the systems {Aj, Aj+I . __ Am j = 2,3.... Hence P (A

Aj3

\

j=2

<_P(AI)

H1(1 -- Tj)• -
completing the proof. In most applications an event Ai is independent of a family {Aj : I E L} if Ai is independent of each A&, I E L. In that case the sets J(i) in Theorem 1.17 are best defined by a dependence graph: a graph F on

1.4 Inequalities

23

(1,2.... m} such that J(i) is precisely F(i), the set of neighbours of i in F. If, in addition, the sets F(i) are fairly small (i.e. F has small maximal degree) then the following variant of Theorem 1.17 may be useful. Theorem 1.18 Let AI,A 2,...,Am be events with dependence graph F. SupposeO< y
7)d'i)-l

--

Jor every i, where d(i) is the degree of i in F. Then (A

Aj) <_ P(Ai)(1 -

< _ y/(17d(l)

If F has maximal degree A > 3 and P(Ai) < (A - 1)A-a/AA then P

(

>0.

Proof The first part follows by imitating the proof of Theorem 1.17, with J(i) = F(i). Note that if j is a neighbour of vertex 1, then j has degree d(j) - 1 in F -- {1}, so with d = d(1) and the notation of the proof of Theorem 1.17, P(AjjAj+j n ... n) Am) ýý_I -- P(Aj)(1 -- 7)-d(j)+1

>_ 1 -- 7.

for 2 _•j _•d + 1. Hence

(A

Aj)

< P(A1 )(1

as claimed. The second part follows by putting

--

)d(1)

=1/A.

< 7/(1

-7),

wj

Note that {(A - 1)/A}Ai- > 1/e if A > 2, so if A(G) = A > 2 and P(A,) <_ 1/(eA) for every i, then P(nm=IAi) > 0. In the original paper of Erd6s and Lovfsz (1975) this was proved under the assumption that P(Ai) :!_1/4A. The Lovftsz Local Lemma enables us to show that certain unlikely events do occur with positive probability, even when this probability is exponentially small. In some applications, mere existence is frequently

24

Probability Theoretic Preliminaries

not sufficient and it is desirable to have a fast algorithm to locate 'rare' objects. This difficult and important problem was solved by Beck (1991), who showed that in some applications the Lemma can be converted into polynomial time sequential algorithms for finding desired 'rare' objects. With this breakthrough, Beck has greatly increased the importance of the Lovwsz Local Lemma to algorithms. Alon (1995) modified Beck's technique and constructed a parallelizable algorithmic version of the LovSsz Local Lemma. Having seen inequalities that guarantee that an event has positive probability even if this probability is exponentially small, we turn to inequalities that can be used to show that certain 'bad' events have exponentially small probabilities. Given a probability triple (2, F, IP), let F0 ( Y-, c ... be increasing sub-a-fields of F.

Let X 0 , X 1,.... be r.vs on Q such that

Xk

is Fk-measurable and

E(Xk+l 1Fk) = Xk. The sequence (Xk) is a martingale. In combinational

applications, more often than not, Q is a finite set and so (Xk) is a finite sequence, each Fk is given by a partition of K into blocks (the atoms of Yk), with the partition of Yk refining that of YFk-l, and Xk = E(XIYk) = E(Xk+llYk) for some r.v. X on Q. Furthermore, usually Y0 is the 6-field (Q, 0) and so X0 = E(Xk) = E(X) for every k. The following Hoeffding-Azuma inequality was proved by Hoeffding (1963) and Azuma (1967). Theorem 1.19 Let (Xk)o be a martingale such that [Xk - Xk-1 I-1.. ... l,for some constants cl,..., c1 . Then, Jor every t Ž0, IP(X

_ X0 + t) 5 exp {-t2

2

Ck,k =

kck}.

In combinatorics, it is often convenient to apply a variant of this inequality. The result below is due to Janson (2001), who proved the inequality with a particularly good constant in the exponent. Theorem 1.20 Let Z 1 ,...,Zj be independent r.vs, with Zk taking values in a set fk. Let f : Q = Q,1 X ... x Ql --* R be a measurable function such that if woc 0 and o' E K2 differ only in their kth coordinates then

f(O) --(o')[ < Ck Jor some positive constants c 1,..

-. Ck.

Then the random variable X

=

1.5 Convergence in Distribution f(Zl ....

25

Z1 ) is such that, for every t >0 ,

IiNX•<EX -t)•:ýexp{~t Y Replacing f by -f, we see that IP(X >_ EX + t) _•exp{-2t 2 //1 c2} holds as well. Frequently, the set-up is even simpler: Q 1, .,k are finite probability spaces with product f, and X : Q - IR is a r.v. that satisfies the Lipschitz condition in Theorem 1.20: !X(Ow) - X(wO')l _
ck

for every t > 0.

1.5 Convergence in Distribution The sieve formulae of the previous section imply a number of results about convergence in distribution. In particular, Corollaries 1.12 and 1.13 have the following immediate consequence. Theorem 1.21 Let X 1, X 2 .... and X be non-negative integer-valued random variables. Suppose lim Er(Xn) = Er(X), r = 0, 1,.... nf-C/

and lim Er(X)rm /r! = 0, m = 0, 1.

r-.x

Then d

Xn --+ X. If X is a Poisson r.v. with mean I, then Er(X) = ýr for every r. Hence, Theorem 1.21 implies the following result. Theorem 1.22 Let 2 = 2(n) be a non-negative bounded function on N. Suppose the non-negative integer valued random variables X 1 ,X 2,... are

26

Probability Theoretic Preliminaries

such that r = 0,1 .....

lim{Er(Xn) - 2r} = 0,

Then d( Xn, PA) ---- O. This simple result will often be used in the main body of the book [see also Bender (1974a) for many other applications and refinements of the method]. We shall also need the analogue of Theorem 1.22 for several sequences of r.vs. The result below is implied by Corollary 1.15. Theorem 1.23 Let 2l = )1(n)......m = 2m(n) be non-negative bounded functions on N. For each n let X, (n), ... , Xm(n) be non-negative integer valued random variables defined on the same space. Suppose.for all rl, ... , rm G Z+ we have lim (E {(Xi (n))

Then X, (n),

... , X.(n)

,, ).

..

- 2.

..

M

0.

are asymptotically independent Poisson random vari-

ables with means '1,..,m, that is lim

=

kl,...,Xm(n)

=

kmn}

(e-A,
-

=0

i=1j

,for all kl,....km e Z+.

We know that if . > 0 is a constant and pn if pqn ---> oc then (Sp, -pn)/

di

-

2, then Sp,n

ppqn -) N(O, 1). Hence (P 2 -2)/

d d

)

P;. Also, d

N(0, 1)

as oo. [This is another special case of the Central Limit Theorem (see Theorem 1.25, below), since by Ex. 5 if X 1, X 2., .. X, are independent Poisson r.vs with mean 1, then E" 1 Xi has Poisson distribution with mean n.] We shall often need the assertion that similar convergence holds for r.vs which are not necessarily Poisson but are close to being Poisson r.vs. Suppose X is a real-valued r.v. with a continuous distribution function F and absolute moments M 1 , M 2 ,....: -A-

Mn=

J

xdF.

If all moments exist and lim sup M 1 /"/n< oc, n----2

27

1.5 Convergence in Distribution

then F is uniquely determined by its moments. In fact, by Carleman's theorem F is uniquely determined by its moments iff E, I Mn-l/n = 0c (see Feller, 1966, vol. II, p. 487). Since the nth absolute moment of N(0,1) is asymptotic to ,/2(n/e)n/ 2 (see Ex. 8), the normal distribution is determined by its moments. This implies that if (Xn) is a sequence of real valued r.vs such that for every fixed r the rth moment of Xn tends to the d

rth moment of N(0,1), then X, -'N(O, 1): The observations above lead us to the following simple result. Theorem 1.24 Let (X,) be a sequence of r.vs and let 2, --+ 0c. Suppose =o(,q5 )

Er(X,,) -

,for all r and s, 1 < r < s. Then S-N(O, 1). Proof Clearly, for every k c N, k

E(X-

)kk/2} =

k

)

ck,rtEr(Xn)..-jn2 '

(1.21)

r=O t=O

where Ck,r,t depends only on k, r and t.

(Yn

Let Y 1, Y2 .... be Poisson r.vs, with Yn having mean An, and set Zn - ýO2n/. Then Z-->N(O, 1) and so k

E (Z,)

n

=

=

k

3>c

n Ck,r,tr~nt/-kl

--- +Mk,

r=0 r=0

where mk is the kth moment of N(0,1). Therefore (1.21) and our assumption on Er(X,) imply that the kth moments of the sequence (Xn -- )Y2n 1/2 also converge to Mik. Convergence to normal distribution holds under rather weak conditions. Here we shall just state two versions of the Central Limit Theorem holding for variable distributions. The following result, which is essentially the best possible, is due to Lindeberg (1922) (see also Feller, 1966, vol. II, p. 256).

28

Probability Theoretic Preliminaries

Theorem 1.25 Let XI,X 2,... be independent random variables with Xk having mean 0, variance or2 > 0 and distributionfunction n Sn

E

tk.

k=l

Suppose n

y 2 dFk(y) -*0

sn-, E-•

(1.22)

[>_!tsý [k=l

•

for every t > 0. Then

E- X1/sn

__'N(O, 1).

k=1

It is not always easy to check whether (1.22, called the Lindeberg condition, is satisfied; often it is more convenient to check the Ljapunov condition, (1.23), given below (see Ex. 22). Corollary 1.26 Suppose (Xk) and (sn) are as in Theorem 1.25, and for some •5>0 n

2 (JXkk ֥i

Z

=

o(sn÷6 ).

(1.23)

k=i

Then the Lindeberg condition (1.22 is satisfied and _ XI/s

- *

N(O, 1).

k~l

In combinatorial applications one usually checks (1.23) with 6 = 1. In estimating joint probability distributions, the following simple result is often useful. Theorem 1.27 Let fl(x1,...,

XN), f 2 (xI, ...

XN) ...

fr(x ..

XN)

negative functions, non-decreasing in each variable xi. Then if X 1. are independent r.vs, we have

be nonXN

It is easily seen that the general case follows from the case r = 2 and XN) by fj(-xI .....- -XN) we see that the same inequality holds if the functions are non-increasing in each variable xi. N = 1 (Ex. 21). By replacing f1 (xl. .

1.5 Convergence in Distribution

29

Theorems 1.22 and 1.24 give rather simple-minded, though very useful, methods for showing approximations by Poisson and normal distributions. In the context of random graphs, a considerably more sophisticated method was used by Barbour (1982). This approach is based on that of C. Stein (1970) for the normal distribution and that of Chen (1975) and Barbour and Eagleson (1982) for the Poisson distribution. The approximation by the Poisson distribution rests on an ingenious observation we shall state as a theorem. For A ( Z+ and A > 0 let X = XAA be the real-valued function on Z+ defined by x(O) = 0 and Ym-lem!{P(A n Cm)

x(m + 1) =

-

P(A)P,(Cm)},

m > 0.

Here Cm = {0, 1.

,

and P;(B) = e-' ZE k/k! kcB

is the probability that a Poisson r.v. with mean ) has its value in B. Theorem 1.28 The Junction x

=

x;.,A

has the following properties:

(i) jxkj -- sup. jx(m)] _<min{1,2•-'/21, (ii) Ax -- supm x(m + 1)-- x(m)] < 2min{1,)- 1 }, (iii) for any non-negative integer valued r.v. Y E{2x(Y + 1) - Yx(Y)} = P(Y G A) - P;(A). Proof For a proof of (i) and (ii) we refer the reader to Barbour and Eagleson (1982). Although the proof of (iii) is even more straightforward, we give it here, since the significance of x is exactly that with the aid of x we can express the difference P(Y e A) - P2 (A) in a pleasant form. Assertion (iii) is equivalent to the relation x(k + 1) - kx(k)

I - P(A), if k E A, 1-P (A), if k 4 A.

(1.24)

To see (1.24) note that if k > 1, then ,x(k + 1) - kx(k)

=

2 ke 2k!{P),(A n Ck) - P2 (A)PA(Ck) - P(A n +P).(A)PA(Ck-1)}

= [PA(A n {k}) - P),(A)P),({k})]/P;,({k}) = P;,(A n {k})/P 2 ({k}) - P),(A).

Ck-1)

Probability Theoretic Preliminaries

30

and for k = 0 we have

2x(1) = eA[P;(A n {0}) - P;.(A)P;.({O})] = PJ(A n {O})/Pý({}) - P(A).

D

Theorem 1.28 may enable us to estimate the total variation distance of a distribution from a Poisson distribution. In this way we may prove a convergence rate for X,,

d -

P;.

Exercises 1.1 1.2 1.3

1.4 1.5

Prove inequalities (1.8), (1.9) and (1.11). Check that O(x) is indeed a density function. Let Y be a geometric r.v. with mean q/p, i.e. let P(Y = k) = qkp, k = 0,1 ..... Prove that E(Y) = q/p,a2 (Y) = q/p 2 and Er(Y) = r!(q/p)r. Show that an exponential r.v. with parameter 2 > 0 has mean 1/2 and variance 1/22. Let (Y,) be a sequence of geometric r.vs with Y,, having mean d

1.61.7 1.8

(1 -- p,)/p,. Show that if p, -+ 0, then pY, -L, where L is an exponential r.v. with mean 1. Let X and Y be independent Poisson r.vs with means 2 and u. Prove that X + Y has Poisson distribution with mean 2 + p. Prove formula (1.15). [See Feller (1966, vol. I, p. 179) for a proof of the case 1= 0.] Prove that the rth absolute moment of N(0, 1) is (r - 1)!! = (r- 1)(r-3)...3 A1 ,2(r/e)r/ 2 if r is even, and (1/ /)2'/ {(r1)/21! - 1/2(r/e)r/ 2 if r is odd. Indeed, 1~

xlrex 2rc 12dx

2

11

j

- 1

reX22d t 11/2(2t)r12e-tdt t(r- 1 )/ 2 e - td t

,[ 2Er/2

-

,-

2r/2E

l)

(r

where t = x 2/2. If r is even, then I(r+1) 2

r--lr• 2-

3

•

2 2r= .. 1 I(r12)

2

x (r

- 1)!

Exercises

31

and if r is odd then O{(r + 1)/2} = {(r - 1)/2}! [See Gradshteyn and Ryzhik (1980, p. 337), Spiegel (1974, p. 268) or Titchmarsh (1964, p. 63.] 1.9-.

Deduce from Ex. 8 that for every natural number n we have

j

2

xne X /Zdx •

(n!)1/ 2 .

1.10 1.11-.

Prove Theorem 1.7. Let X be a non-negative integer-valued r.v.. Show that E(X 2 )> E(X) and P(X = 0) > 1 - E(X).

1.12

Given ma > 0, m; > 0 and e > 0, construct a non-negative integer-valued r.v. X such that E(X) = mi, E(X 2 ) > mzandP(X = 0)

1 - m, +

[This shows that a large variance cannot improve the trivial lower bound on P(X = 0) given in Ex. 10.] 1.13

1.14

Apply induction on the number of sets to show that inequalities (1.18) and po = 1 characterize the functions P1 defined after Corollary 1.9. (R~nyi, 1970a.) Let g = g(A,,...,An) = ZN-l E'= 1 ci,Jp(Fi)P(Fj), where the ci,j are real constants and F1 ,...,FN are Boolean functions of the variable events A 1 .... A,. Suppose g(A,,....A,) = 0 whenever each Ai has probability 0 or 1. Prove that g(A1 ..... A,) > 0 holds in every probability space and for all choices of the Ai's iff it holds for all choices of the events in the space {xl, x2 }, P(x,) P(x 2 ) = 1. (Galambos and R~nyi, 1968.)

1.15

=

Let L be the lattice of all subsets of [n] = {1,2_..n} and let 29 be the set of probability measures on [n] such that no point gets measure 0. Then every P c -9. can be considered as a lattice homomorphism L -- [0, 1], A - P(A). Finally, let us say that two maps 0, ip : L --+ [0, 1] are homotopic if O(A) < O(B) iff W(A) < W(B). Is it true that a lattice homomorphism t homotopic to Op for some p E -9n iff fo(A) < r(B)iff(A\B) < AE(B\A)

for all A, B

E: L ?

: L -,

[0, 1] is

32 1.16

Probability Theoretic Preliminaries Note that for m = In and

1-

I the second inequality in

(1.5) is of the form (n)

A-An itrnun.

<

Prove that for m = lin > pn we have

P(Sn,p •ý M) =

k=n

pkqfl-k <_(l)"(l)" (n~)

[See Chernoff (1952) and Peterson (1961).] 1.17

1.18

1.19

Let Ai and F(i) be as in Theorem 1.18 and suppose IF(i)[I

1 for

every i. Show that if P(Ai) < 1/2 for every i, then P(flm1 1 Ai) > 0. [Note that one may have P(Ai) = for every i and p(nrlL Ai) 0.] Give an example of a system {Ai}l' such that, with the notation of Theorems 1.17 and 1.18, 1F(i)I = d for every i but there is no dependence graph with maximal degree 1. [Hint: Let m = 2k + 1 > 3 and let Ai,A 2,..., Am be such that any m - 1 of the Aj's form an independent system, but the entire system is not independent.] Show that if A,A 2 ... ,Am are as in Theorem 1.17, 0 < 6iP(Ai) < 0.69 and log5i > Z {6jP(Aj) + 62p(Aj)2 jEJ(i)

i= 1.

m, then nAj) m

_ f(1--

6jP(Aj)) > 0.

=1j=1

[Set Ti = 6iP(Ai) and apply (1.10) to show that the conditions of

1.20

Theorem 1.17 are satisfied.] Show that with the assumptions of Theorem 1.18, one has

P nAj 1.21

0

Let fI and f2 be non-negative and non-decreasing functions, and

let XI, X 2, be independent, identically distributed r.vs. Note that E[{fi(Xi) -fi(X

2 )}{f 2 (X1

) -f

2

(X 2 )}] Ž0

Exercises

33

and so E{(fl(X 1 )f 2 (XI)} Ž E{fl(XI)}E{f

1.22

2 (XI)}.

Show that this implies Theorem 1.27. Prove that the Ljapunov conditions (1.23) imply the Lindeberg condition (1.22).

2 Models of Random Graphs

Throughout this book we shall concentrate on labelled graphs. This is partly because they are easier to handle and also because, as we shall see in Chapter IX, most assertions about labelled graphs can be carried over to unlabelled graphs without the slightest difficulty. For the sake of convenience in most cases we consider graphs with n vertices and take V = {1,2,..., n} to be the vertex set. The set of all such graphs will be denoted by Wn. In the first section we introduce the most frequently encountered probability spaces (models) of random graphs. The following two sections contain basic properties of the models. The last section is devoted to the model of regular graphs. Unlike the case of the other models, a fair amount of effort is needed before one can prove even the simplest results about this natural and useful probability space.

2.1 The Basic Models The two most frequently occurring models are 9i(n, M) and W{n, P(edge) = p}. The first consists of all graphs with vertex set V ={1,2,..., n} having M edges, in which the graphs have the same probability. Thus with the notations N = ('), which we shall keep throughout the book, 0 <_ M < N,W(n,M) has ('t) elements and every element oc1. Almost always M is a function of n: curs with probability () M = M(n). If we want to emphasize that the probability and expectation are taken in W{n,M(n)}, then we shall write PM(oQ/) = PM(f,)(.) and EM(X) = EMtn)(X) instead of P(c/) and E(sI). In the model 1{n,P(edge) = p} we have 0 < p < 1, and the model

consists of all graphs with vertex set V 34

=

{1, 2,..., n} in which the edges

2.1 The Basic Models

35

are chosen independently and with probability p. In other words, if Go is a graph with vertex set V and it has m edges, then P({Go}) = P(G = Go)

pmqNm--.

Here, as everywhere else in the book, q stands for 1 - p. On the other hand, if we want to emphasize that the probability and expectation are taken in ý{n, P(edge) = p}, then we write Pp and Ep instead of P and E. This will be especially important if, as is often the case, p = p(n) is a function of n. Note that f,{n, P(edge) =} is exactly Sn with any two graphs being equiprobable. When there is no danger of confusion, we shall use natural abbreviations. Often we write SM,S(M) or S{M(n)} for S{n,M(n)} and S(n,p),pAj(p) or S{p(n)} for S{n,P(edge) = p(n)}. Furthermore, Gp and GM stand for random graphs from ý#(n,p) and S(n, M). Thus P(Gp is disconnected) is the probability that a graph in S6{n,P(edge) = p} is disconnected. If there is some danger of ambiguity, then we write Gn,p and Gn,m, emphasizing that our graphs have n vertices. Finally, the expression 'random graph' is frequently abbreviated to 'r.g.'. Occasionally we shall comment on a natural refinement of W{n, P(edge) = p}. This is the model W{n,(pij)}; where 0 _
36

Models of Random Graphs

A property Q will be said to be monotone increasing or simply monotone if whenever G E Q and G ( H then also H E Q. Thus the property of containing a certain subgraph, for instance a triangle, a complete graph of order five or a Hamilton cycle, is monotone increasing. We call Q convex if F c G c H and F c Q,H c Q imply that G e Q. If Q is any property, then PM(Q) has the natural meaning: it is the probability that a graph of N(n, M) belongs to Q. Of course, Pp(Q) is defined analogously. Let Q,, be a model of random graphs of order n. [Thus we usually have 9, = N{n,M(n)} or Qn = <(n,p).] We shall say that almost every (a.e.) graph in 92n has a certain property Q if P(Q) -- 1 as n -. oo. Occasionally we shall write almost all (a.a.) instead of almost every. It is perhaps worth noting that the assertion 'a.e. G e ,?6(n, 2) has Q' is the same as 'the proportion of all labelled graphs of order n that have Q tends to 1 as n -> o'.

Let us prove the heuristically obvious fact that a monotone increasing property is the more likely to occur the more edges we have or are likely to have. Theorem 2.1 Suppose Q is a monotone increasing property, 0 •!_M, < M 2 <_ N and 0 <• pI < P2 < 1. Then PM,(Q) <--PM,(Q) and Pp,(Q) < Pp 2 (Q). Proof (i) Let us select the M2 edges one by one. The resulting graph will certainly have property Q if the graph formed by the first M1 edges only has the property. (ii) Put p = (P2

- P1)/(1

- PI). Choose independently G1 E W(n, pl) and

G G '?(n,p) and set G2 = G1 U G. Then the edges of G2 have been selected independently and with probability p1 + P - PIP = P2, so G 2 is exactly an element of W(n, p2). As Q is monotone increasing, if G1 has Q so does G 2 . Hence PPI(Q) •- PP2 (Q), as claimed. Ll Note that the argument above shows that if Q is a nontrivial monotone increasing property then PP2(Q) ->PPt(Q) + {1 - Pp,(Q)}Pp(Q) > Pp,(Q) + Ppt(En)Pp(Kn) > Ppt(Q). In Theorem 2.1 it is irrelevant that we are dealing with graph properties. Let X be a set consisting of N elements and let Q be a monotone increasing property of subsets of X, i.e. let Q •(X) be such that if A e Q and Y A c B ( X then B c Q. For 0 < p < 1 let Pp(Q) be the probability that

2.1 The Basic Models

37

if the elements of X are selected independently and with probability p then the set formed by the chosen elements belongs to Q. Thus Pp(Q) = E PIAI(I (

p)N-IAI.

ACQ

The second part of Theorem 2.1 says that Pp(Q) is a monotone increasing function of p. Let us give another proof of this fact. This proof is somewhat pedestrian, but fairly informative. We may assume that Q is non-trivial, i.e. 0 ý Q and X c Q. Set Hk={A

:A cQ, AI

=

k} and hk

lHkl

=

Then h0 = 0, hN = 1, N

pp(Q)

=

E

hkpk(l

_ p)N-k,

k=O

and so for 0

dPp(Q)/dp

=

=--

N-1

Z khkpk-(1 - p)N-k

Z(N

k=1 N

k=1

Zpk-l1 - p)N-k{khk - (N -- k

- k)hkpk(1

p)f-k

+ 1)hkl}

k=1

Now each set A e Hk-+ is contained in N - k + 1 sets B E Hk and each set B e Hk contains at most k sets A e Hk-l. Hence khk >_(N - k + 1)hk-1, so dPp(Q)/dp > 0, showing that Pp(Q) is an increasing function of p. We shall find it extremely useful that in most investigations the models N(n, M) and N{n,P(edge) = p} are practically interchangeable, provided M is close to pN. The first two parts of the next theorem show that this is the case when we deal with properties holding for almost all graphs. The third part, essentially from Pittel (1982), allows us to replace N(n, M) by V(n,p), p = M/N, provided Pp(Q) = o(M 1/ 2 ). Theorem 2.2 (i) Let Q be any property and suppose pqN --+ oo. Then the following two assertions are equivalent. (a) Almost every graph in S(n, p) has Q.

38

Models of Random Graphs

(b) Given x > 0 and

P

> 0, if n is sufficiently large, there are

1 _> (1 - E)2x(pqN)1 / 2 integers M1 , M 2 ,..., , 1/ 2 pN - x(pqN) 1/ 2 < M, < M 2 < ... < M < pN + x(pqN) such that PM,(Q) > 1 - 8.for every i, i= 1.... 1. (ii) If Q is a convex property and pqN --* o0, then almost every graph in %(n,p) has Q iff Jor every .fixed x a.e. graph in 1(n, M) has Q, where M = [pN + x(pqN)'/2]. (iii) If Q is any property and 0 < p = M/N < 1 then PM(Q) <-- Pp(Q)el/ 6M(27EpqN)l/

2

< 3M 1/2pp(Q).

Proof By the DeMoivre-Laplace theorem (Chapter 1, Theorem 2.6), for every fixed x > 0 Pp{le(G) - pNI < x(pqN)'1 2 }

-

tD(x) - 1D(-x).

Since also 1

Pp{e(G) = M} = b(M;N,p) < (pqN) '/2 for every M, the equivalence of (a) and (b) follows. To see (ii), recall Theorem 2.1 and apply (i). Finally, (iii) follows since by inequality (2.5) of Chapter 1 we have

"PP(Q)

=

Z (N

(N)pmqN-mpm(Q) pmqN-MpM(Q)

1 1 > PM(Q)[e /6M(27rpqN) /2]-l.

This result shows that if we know PM(Q) with a fair accuracy for every M close to pN, then we know Pp(Q) with a comparable accuracy. The converse is far from being true for a general property. As a trivial example, let Q be the property that the graph has an even number of edges. Then Pp(Q) ,- 1/2 whenever pqN --+ oc. On the other hand, if M = 2[pN/2] then 0 _
2.1 The Basic Models

39

be discarded. For example, if Q is the property that the number of edges is not a square, then if pqN -+ oc then Pp(Q) -- 1 but PM(Q) = 0 for M = [(pN)'/ 2 J2 . The second useful connection between Sp and %Mis

only a little less trivial, though we state it as a theorem. Theorem 2.3 Suppose a > 0 and the function X " Nn __, [0, a] is such that X(G) < X(H) wherever G c H. Assume, furthermore, that 0 < pt(n) < p2(n) < 1, M(n) c N, limp qiN = limp 2 q 2N = oo and lim(M -- piN)/(piqiN)11 2 = lim(p2N - M)/(p 2 q 2 N)1/ 2 = 00, n ni

where qi = 1 - pi, i - 1, 2. Then E,,(X) + o(1) _<EM(X) < Ep2(X) + o(1). Finally, if we are interested in EM=0 EM(X), the sum of expectations of a r.v. X, then we do not need any assumptions on X to replace the model #(n, M) by W(n, p). Theorem 2.4 If X : Wn --* R is any function, then N

1

S EM(X) = (N + 1)1

,10

M=0

E(X)dp.

Proof Since N

EP(M

(N) p MqN-~MEm(X), M=0

we have EM(X)

Ep(X)dp =

EM(X).

1

pM(1-_ P)N-Mdp =

M=

+

M=0

The second equality above is based on the identity 1

[

xM(1M

_X)N-Mdx = B(M +II 1,N-M +II 1)-T / A N -

1

NN)+1

where B(v,jy) is the beta function (see Gradshteyn and Ryzhik, 1980, p. 950). The identity is easily proved by partial integration. LI

40

Models of Random Graphs

Erd6s and R~nyi (1959, 1960, 1961a) discovered the important fact that most monotone properties appear rather suddenly: for some M = M(n) almost no GM has Q while for 'slightly' larger M almost every GM has Q. Given a monotone increasing property a function M*(n) is said to be a threshold.function for Q if M(n)/M*(n)

-*

0 implies that almost no GM has Q,

and M(n)/M*(n) -+ oo implies that almost every GM has Q. A priori there is no reason why a property should have a threshold function though this follows from an observation of BollobSis and Thomason (1987), noting that, with the natural definition, every monotone property of subsets of a set has a threshold function. Considerably deeper results were proved by Friedgut (1999): he gave necessary and sufficient conditions for the existence of sharp thresholds for monotone graph properties. When we study a specific graph property Q, we do not rely on general results, and often succeed in doing more thick merely identifying a threshold function: for many a property Q we shall determine not just the threshold function but an exact probability distribution: for every x,0 < x < 1, we shall find a function Mx(n) such that P(GMA as n ---> o.

has Q)

-*

x

It is clear that threshold functions are not unique. However, threshold functions are unique within factors m(n), 0 < lim m(n) <_lim m(n) < co, that is if M, is threshold function of Q then M; is also a threshold function of Q iff M; = O(M,) and M* = O(M2). In this sense we shall speak of the threshold function of a property. As mentioned earlier, the nearer our r.vs are to being independent, the easier it is to handle them. This is the reason why Np is more pleasant to work with than 5m. In Sp the edges are chosen independently, while in the model NM the choice of an edge does have a (fortunately small) effect on the choice of another edge. Another model with good independence properties is <6(n;k-out) = Wk-out. A random graph Gk-out is constructed as follows. The vertex set is V = {1,2,...,n}. For each vertex x c V select k other vertices, all (n k) choices being equiprobable, and all choices being independent, and for each selected vertex y direct an edge from x to y. Let G = Gk-out be the random directed graph constructed in this way, with •

probability space formed by them. Equivalently,

;k-out

=

•k-out the

is the probability

2.1 The Basic Models

41

space of all directed graphs with vertex set V in which every vertex has outdegree k. It is fascinating to note that ! 6k-out is mentioned in the Scottish Book: in 1935 Ulam asked for the probability of Gk-out being connected (see Mauldin, 1979; p. 107, Problem 2.38). For G E i let Gk-out = 0(G) be the random graph with vertex set V and edge set {xy: at least one of iy and jx is an edge of G}, Wk-out = 0(G) : G E <ýk-out. The probability of a graph H in %k-out is the probability of the set of graphs 0- 1(H) in ;k-out. Note that every Gk-out has at most kn edges but every vertex of it has degree at least k. As we shall see in Chapter 3, this is in sharp contrast with the graphs GM, for a.e. GM has isolated vertices unless M is not much smaller than (n/2) log n. The model W(n;p 1 ,p172 . -. Pk) is also obtained by identifying edges obtained in different ways. The graphs of this model are obtained by first selecting edges of colour 1 with probability Pl, then edges of colour 2 with probability P2, etc. This gives us a multigraph (graph with multiple edges) whose edges are coloured 1,2,..., k. To obtain our r.g. we just forget the colours of the edges and identify multiple edges. Of course, the model we get in this way is nothing but W(n,p), where p = 1- _H•(1 -pi). However, by selecting the edges in several stages we can work with independent graphs: the graph formed by the edges colour i is independent of the graph formed by the edges of colour j, j = i. This model was used in the second part of the proof of Theorem 2.1. There are many other models of random graphs. By giving all graphs in a finite set the same probability we obtain a probability space. In this way we have random k-regular graphs, random trees, random forests, random planar graphs, etc. Of these, only random regular graphs will be investigated in detail. As regular graphs are not easily generated, we devote an entire section (§4) to the introduction of random regular graphs. Let us conclude this section with two models which, in some sense, are refinements of %(n, p) and S(n, M). Let 0 < p < 1 be fixed. The model ý§(N, p), studied in Bollobfis and Erdds (1976), consists of all graphs with vertex set N, whose edges are chosen independently and with probability p. In other words a r.g. G e S(N,p) is a collection (Xij) = {Xij : 1 i < j} of independent r.vs with P(Xij = 1) = p and P(Xij = 0) = q: a pair ij is an edge of G iff Xi = 1. Clearly G, = G[1,2,...,n], the subgraph of G spanned by 1,2,..., n, is exactly a r.g. Gp on V = { 1, 2,..., n}. Note that the space W(N, p) has uncountably many points and depends

42

Models of Random Graphs

only on p. The definition of 'almost every' is then just the standard definition in measure theory: almost every (a.e.) G E W(N,p) is said to have a property Q if P(G has Q) = 1. Note that if a.e. G is such that for some n(G) E N every G, n > n(G), has property Q, then a.e. Gp has Q. On the other hand, the converse is far from being true. Thus assertions concerning a.e. graph in !(N, p) tend to be stronger than assertions about a.e. Gp. A random graph process on V = {1, 2,..., n}, or simply a graph process, is a Markov chain G = (Gt)', whose states are graphs on V. The process starts with the empty graph and for 1 •< t < (•) the graph G, is obtained from G,_1 by the addition of an edge, all new edges being equiprobable. Then Gt has exactly t edges, so for t = (2) we have Gt = K'. Fort > (n) we take Gt = K" as well. A slightly different approach to random graph processes goes as follows. A graph process is a sequence (Gt)4 = 0 such that (i) each G, is a graph on V, (ii) Gt has t edges for t = 0, 1,...,N and (iii) Go = G, c. Let 4 be the set of all N! graph processes. Turn W_into a probability space by giving all members of it the same probability and write G for random elements of i. The term 'almost every' is used in its familiar meaning: almost every graph process G is said to have property Q if the probability that 6 has Q tends to 1 as n -* c. Furthermore, we call Gt the state of the process G = )N at time t. The map -- . 1(n, M), defined by G = (G,)N -4 GM, is measure preserving so the set of graphs obtained at time t = M can be identified with N(n, M). Hence no contradiction will arise from the fact that GM stands for two different objects: a random graph from !§(n,M) and the state of a graph process 6 at time M. Intuitively, G is an organism which develops by acquiring more and more edges in a random fashion. One of the main aims of the theory of random graphs is to determine when a given property is likely to appear. As mentioned earlier, Erd6s and R~nyi were the first to show that most monotone properties appear rather suddenly. In rather vague terms, a threshold function is a critical time, before which the property is unlikely and after which it is very likely. Suppose Q is a monotone property of graphs. The time z at which Q appears is the hitting time of Q: 'r= rcQ= TQ(G) = mint

>0

: Gt has

Q}.

2.2 Properties of Almost All Graphs

43

Thus M*(n) is the threshold function of Q if whenever wo(n) --+ oo, the hitting time is almost surely between M*(n)/lo(n) and M*(n)co(n):

P{M*(n)/wo(n) < TQ(n) < M*(n)wo(n)} ---* 1. Hitting times enable us to relate properties of r.gs more precisely than the model <(n, M). For example, let Q, be the property of having no isolated vertex and Q2 the property of being connected. Then zQ,(G) < zQ 2(G), i.e. Q2 cannot occur before Q, since a connected graph (of order at least 2) has no isolated vertex. As we shall see in Chapter 7, a.e. GM has Qi (i = 1 or 2) iff M = (n/2){logn + wo(n)}, where co(n) -* oc. Thus a.e. GM is connected iff a.e. GM has minimum degree at least 1. However, considerably more is true. The main obstruction to connectedness is the existence of an isolated vertex: a.e. graph process G is such that [Q, = TQ2, i.e. the first edge incident with the last isolated vertex also makes the graph connected.

2.2 Properties of Almost All Graphs If M is neither too small nor too close to N then for every fixed graph H a.e. GM has the rather pleasant property that the graph H can be embedded in it vertex by vertex. To be more precise, given graphs F = H, in a large range of M a.e. GM is such that if GM has an induced subgraph isomorphic to F then GM also has an induced subgraph H* such that H* : F*, H* _- H and the isomorphism H* ý-- H is an extension of F* -- F. This is an immediate consequence of the following result. Let us say that a graph G has property Pk if whenever W 1 , W2 are disjoint sets of at most k vertices each then there is a vertex z E V(G) - W, U W2 joined to every vertex in W 1 and none in W 2 . Clearly Pk+1 implies Pk. Theorem 2.5 Suppose M = M(n) and p = p(n) are such that for every E > 0 we have Mn-2+

-

pn'

---

oc and (N - M)n-2+e oo and (1 -p)n'

-+

O C,

oo.

(2.1) (2.2)

Then for every.fixed k E N a.e. GM has Pk and a.e. Gp has Pk. Proof Although Pk is not a convex property, we can get away with considering only the model S(n,p). Indeed, M(n) satisfies (2.1) if and only if p = M/N satisfies (2.2). Therefore by Theorem 2.2(iii) it suffices

44

Models of Random Graphs

to show that if p satisfies (2.2) then PP(]Pk) = o(n-1). This is true with plenty to spare. Note that if iG1 > 2k, then G has Pk if and only if a vertex z can be found for all pairs W 1, W 2 with IWI = IW21 = k. Suppose p satisfies (2.2). Write Pp(W 1 , W2 ;z) for the probability that a fixed vertex z 0 W1 U W2 will not do for a given pair (W1, W2),I W1 = IW21 = k, and let Pp(W 1 , W2 ) be the probability that no vertex z will do for (W1, W2). Then P(WI, W 2 ;z) =

1--pkqk,

P(W1, W2) = (1 - pkqk)n- 2 k <_ exp{-(n - 2k)pkqk}I_ exp(-n

1 2

/ )

and so Pp(]Pk) -- n 2k exp(--n1/ 2 ) = o(n- 1).

D

The metatheory of graph theory could hardly be simpler. In the theory of graphs there is only one relation, adjacency, satisfying the condition xRy -+ yRx and ]xRx, where xRy means that the vertex x is adjacent to the vertex y. Consequently, first order sentences (sentences involving = I V, A, 1, V,1, -+ and R) are particularly simple. As is shown by the next

result, which is a slight extension of a theorem of Fagin (1976), for a large range of M and p first-order sentences are not too interesting when applied to graphs in N(n,M) and W(n,p). Theorem 2.6 Suppose M and p satisfy (2.1) and (2.2), and Q is a property of graphs given by a first-order sentence. Then either Q holds for a.e. graph in ,?•(n, M) and W(n, p) or it fails.for a.e. graph in W(n, M) and W(n, p). Proof We claim that there is a unique graph (up to isomorphism) with a countable vertex set which has Pk for every k. An example of such a graph is Go = (N,E) with E = {ij : i < j,pilj}, where P1 ,p 22.... is an enumeration of the primes. The uniqueness follows from a standard 'back and forth' argument. Suppose G and H have Pk for every k, V(G) = {gl, 2,...} and V(H) = {h1 ,h 2 .... }. Define 'partial isomorphisms' a, {gi,..., gi,, -} {hi ..... hi,} inductively as follows. Let gil = gl and hi = hi. If n > 2 is even, pick i,+1 to be the least i > 1 not in {i. . in}. Since H has P, there is a vertex hj,+, ý {hj1,....,hj} such that for all s, 1 _<s < n, we have hj1,,Rhj, iff giRg•,. If n > 2 is odd, reverse the role of G and H above. The union of the partial isomorphisms is clearly an isomorphism between G and H.

2.2 Properties of Almost All Graphs

45

Since the theory of graphs together with all Pk's has no finite models and all countable models are isomorphic, the theory is complete (see Vaught, 1954; Gaifman, 1964). Hence if Q is any first-order sentence then either Q or ] Q is implied by some set of Pk's and so by some Pk, say by Pk,. In particular, either P(G has Q) > P(G has Pk,) or P(G has ]Q) > P(G has Pk,). By Theorem 2.5, a.e. graph in our range has Pk,, so the

proof is complete.

LW

For monotone properties Theorem 2.6 can be restated in terms of hitting times as follows: if Q is a monotone property given by a firstorder sentence then for almost no graph process Q does M = cQ(6) satisfy (2.1). The moral of Theorem 2.6 is that if we wish to study a property of random graphs given by a first-order sentence, then we have to do it in models that do not satisfy (2.1) and (2.2). In fact, many graph properties one studies are not given by first-order sentences: 'G is bipartite', 'G has chromatic number at least k', 'G is Hamiltonian', 'G contains [n 1/ 2] vertices dominating all vertices', etc. Theorems 2.5 and 2.6 are essentially best possible, as is shown by the following result. Theorem 2.7 Given r > 0 choose k E N such that ke > 1 and define p = p(n) > 0 by nk epkn = k!

2

Let Q be the property that for every set of k vertices there is a vertex joined to all of them. Then 0 < p < e if n is sufficiently large and _5 1S< limP(Gp has Q) < limP(Gp has Q) < 5

Proof By definition

so p < g if n is sufficiently large. Denote by X = X(Gp) the number of k-tuples of vertices of Gp for which no vertex can be found which is joined to all of them. Then E(X) =

n) (1

- pk)n-k

_

nke -pknk! ____ 2'

since the probability that a vertex x will not do for a k-tuple W is P{xy E E(Gp) for some y G W} = 1-P{xy ý E(Gp) for all y c W}

= 1-pk.

46

Models of Random Graphs

Now let us calculate E2 (X). There are (n)(1)(n-1) ordered pairs of k - 1. The probability that a vertex k-tuples sharing 1vertices, I = 0, 1 x is not good for either of these k-tuples is I - p I + pI(1I_ pk-1)2 = 1-22pk + p~k-1. For I = 0 this is (1 - pk)2 and for 1 > 1 this probability is at most (1 _ pk)2(1 + p2k-1).

Hence

k -

( n (k

n -- k)

pk)Zn-4k(1

p2k- I)n-2k

k

(X)2 +

=

O(n2k-Ie-2pkn), /=1

since (1+

p2k-I)n •

e

'k

1

-

1+

{(logn)kl/n}I/k.

Consequently, E2(X) = E(X)2 {1

O(1/n)}

*

4

By Corollary 1.11 (or the first part of Theorem 1.16) and the trivial inequality P(X = 0) > 1 - E(X), we have I -1- E(X) < P(X = 0)• 1- E(X) +-E 2(X) 8 This proves our theorem, since Gp has Q iff X(GP) = 0.

F]

The necessity of conditions (2.1) and (2.2) in Theorem 2.5 is also an easy consequence of some results in Chapter 4 (see Ex. 4 of Chapter 4).

2.3 Large Subsets of Vertices The main reason why random graphs so often provide examples of graphs which are not easily constructed is that in a loose sense a random graph is an almost regular graph with surprisingly strong homogeneity properties. To be a little more precise, in large ranges of p and M most r.gs GP and GM are such that their vertices have about the same degrees and all not too small sets of vertices behave in a very similar way.

47

2.3 Large Subsets of Vertices

This similarity in the behaviour of sets of vertices will be discussed in this section, and the next entire chapter will be devoted to the degree sequence. Consider a graph G E •p. For brevity we write e(U) = e(G[U]) for the number of edges of G in the subgraph G[U] spanned by a set U ( V, that is for the number of edges joining vertices of U. Similarly we write e(U, W) for the number of U- W edges, where U and W are disjoint sets of vertices. Clearly the expected value of e(U) is p (2) and the expected value of e(U, W) is puw, where u and w denote the number of vertices in U and W, respectively. An important aspect of the similarity of not too small sets of vertices is that almost every graph is such that e(U) is close to p (2) and e(U, W) is close to puw whenever U and W are sufficiently large. Theorem 2.8 Let 0 < p = p(n) < . Then a.e. G c W is such that if U is any set of u > ( 2 52 /p) log n vertices, then

e(U) -p

(u)

< ( 7Plog n) 1/2

(~.(2.3)

Proof Let U be a fixed set of u > (2 52 /p) log n vertices. Then e(U) has binomial distribution with parameters (2) and p. Put r = {(7 log n)/pu}1 / 2 . Clearly by assumption e < 1/6 and

u>

E-2 =

P (U)2

pu 7 log n'

Consequently by Corollary 1.4 we have, for sufficiently large n,

pe( U) -

p

(u)

Ž

p

(u) ) (u) < 2e2p


(2.4)

The second inequality in (4) holds because, if n is large enough, 7 6q

u+3 u-

I

Since there are (n) < n"/u! sets of u vertices, the probability that some set U with u > u 0 = [( 2 5 2 /p) logn] elements violates the inequality of the theorem is at most

Z

-U>UOnu

2<

n

48

Models of Random Graphs

As inequality (2.3) defines a convex property, by Theorem 2.2 the result above holds for §M as well, where M = p (n). Note that Theorem 2.8 concerns only graphs of fairly many edges, for (2.3) holds vacuously unless u < n and so p > (144/n)logn,pn 2 /2 > 72nlog n. In fact, large sets of graphs of much smaller size also behave rather regularly. We state the result for WM. Theorem 2.9 Suppose M = M(n) < n2 /4 and uo = uo(n) < n/2 satisfy Muo/{n 2 log(n/uo)}

oo.

-

Then for every r > 0 a.e. GM is such that every set U of u > uo vertices satisfies le(U) - M(u/n) 2 1 < ,M(u/n)2 .

(2.5)

Proof For a fixed set U of u vertices e(U) has hypergeometric distribution with parameters (n) , (u) and M. By inequality (2.12) of Chapter 1 we could approximate this distribution by b {k; M, (u) (n) } and then could apply Theorem 1.7 to prove (2.5). However, it is easier to deal with the model WP and then rely on Theorem 2.2 to translate the result back to ýVM. Inequality (2.5) defines a convex property, so by Theorem 2.2 it suffices to prove (2.5) for the model Wp, where p = M/ (n). By assumption, pu2 -, oo and we may assume that e < 1/10. Then by Theorem 1.7 for every fixed set U, IUI = u > uO, we have P ( e(U) -p We have

(•)

for some set U,

I

(u)

> 2p

(u))

<e-2PU2 /7.

(2.6)

< (en/u)u choices for U, so the probability that (2.6) fails

IU1 > uo, is at most

exp[u{--e2 pu/7 + 1 + log(n/u)}]

u>uo

•

5 exp[u{--2pu°/7 + I + log(n/uo)}]

= o(l),

Id>U0

for -9 2puo/7 + log(n/uo) < -(E2 /7)Muo/n

2

+ log(n/uo) ---- -oo.

This proves our theorem, since

implies (2.5).

E

2.3 Large Subsets of Vertices

49

Corollary 2.10 If M/n --+ c and E > 0 then a.e. GM is such that 2 2 le(U) - Mu /n ] <

2 eMu /n

2

for every set U of at least En vertices. The corresponding results concerning e(U, W) can be proved analogously to Theorems 2.8 and 2.9. The simple proofs are left to the reader. Theorem 2.11 Let 0 < p = p(n) <_ 1/2. Then a.e. Gp is such that le(U, W) - puwl < (7plogn) 1/2 uw, whenever U and W are disjoint sets of vertices satisfying (2 52 /p) log n < u = 1uI _•w = IWI. Theorem 2.12 Suppose M and uo are as in Theorem 2.9. Then for every e > 0 a.e. GM is such that

le(U, W) - 2Muw/n21 < EMuw/n 2 , whenever U and W are disjoint sets of vertices satisfying uo < u = IU < w = W1. Corollary 2.13 If M/n -+ oo and e > 0 then a.e. GM is such that le(U, W) - 2Muw/n 2 1 < EMuw/n 2 , whenever U and W are disjoint sets of vertices satisfying en w = W1.

•_

u = IU <

The next two results, essentially from Bollobfis and Thomason (1982), are more technical. They also express the fact that in most graphs all large sets of vertices behave similarly. Theorem 2.14 Let 0 < E < 1 be a positive constant, p = p(n) • and suppose wo = wo(n) >_ [6 log n/(r 2p)]. Then a.e. Gp is such that if W = V and IW1 = w _ wo then Zw = {z G V - W : JF(z) n W1 - pwl >_rpw} satisfies IZwJ <

12logn/{e 2p}.

Models of Random Graphs

50

Proof We may assume that wo < n and so pn

-*

oo. For fixed w > wo, W

and x c V - W Corollary 1.4 implies that

P(x E Zw) <_ 2e-'pwv, where q = 12/3. Hence P(IZwl Ž z) < (2n)ze-qPwz < epz

Therefore, if z > 4(log n)/ilp, the probability that some set W of order w exists which fails to satisfy the conclusions of the theorem is at most nWe qpwz/

3

/3

Thus a graph fails the conclusions of the theorem with probability at El most Ew_ niv/ 3 = o(1). Theorem 2.15 Suppose 6 = 6(n) and C = C(n) satisfy 6pn > 3logn.C > 3 log(e/3) and C6n --+ oo. Then a.e. Gp is such thatJorevery U ( V, UI u = [C/p] the set

=

{x• V--U :F(x) fU=0}

T=

has at most 3 n elements. Proof For fixed U and x c V - U we have P(x G Tu) =(1- p)u <_e-PU < e-C. Hence, for a fixed set U, 6n_< • ( 6n n) P (IT,P(I•! I ýn)

Finally, since there are at most

6 e-C 6 nf< (e/6)6neC n <•e2C~n/3.

n[CIPl

choices for U,

P(ITul > 6n for some U) < nc/Plee-2C6n/3

•exp

(4C

2 logn--C6n) •exp

-

1

C6n

=o(1).

2.4 Random Regular Graphs Suppose the natural numbers n and r = r(n) are such that 3 < r < n and rn = 2m is even. Denote by IN(qr-reg) = Ir-reg the set of r-regular graphs. By our assumption on r, the set ",-reg is not empty. We turn Sr-reg into a probability space by giving all elements the same probability, and call an element of this probability space, 'r-reg, a random r-regulargraph.

2.4 Random Regular Graphs

51

Then every graph invariant becomes a r.v. on ýýr-reg and the expression 'a.e. r-regular graph has property Q' needs no explanation. Before getting down to the study of r-regular graphs, a basic question has to be answered. About how many r-regular graphs of order n are there? In other words, what is the asymptotic magnitude of Lr = Lr(n) = VSr-regj? The reason why the models 1(n, M) and V(n, p) have rather long and distinguished histories while the model Sr-reg is only a newcomer is exactly that such an asymptotic formula is not too easily come by. Read (1959, 1960) obtained an exact formula for Lr, based on the enumeration theorem of P61ya (1937) [see Bollobfis (1979a, chapter 8) for a more recent presentation of it]. Unfortunately, this formula is not easily penetrated. In particular, it seems that only for r _<3 can it be used to find the asymptotic value of L, (see Harary and Palmer, 1973, p. 175). Bender and Canfield (1978) gave an asymptotic formula for Lr and, more generally, for the number of labelled graphs with given degree sequences. Earlier, Erd6s and Kaplansky (1946), O'Neil (1969), B~kbssy, B~k~ssy and Koml6s (1972) and Bender (1974b) had studied similar questions concerning matrices and bipartite graphs. Bollobis (1979b, 1980b) gave a simple model for the set of labelled regular graphs, which not only implies a proof of the asymptotic formula, but also makes the study of random regular graphs fairly accessible. The aim of this section is to prove the asymptotic formula via this model. The key result is the following. Theorem 2.16 Suppose A is a fixed natural number, n

A Ž!dl Ž! d 2

d,n > 1Zdi 1Ž

...

= 2m

is even, 2m - n --* oo and Go is a graph of maximum degree at most A

with vertex set V = {1,2,...,n}. Denote by Y(d; GO) the set of graphs with vertex set V which do not share an edge with Go and whose degree sequence is exactly (di)7 d(i) = di, i = 1.... ,n. Then I

Y(d ; G o)[

e-

2 )12-'

/4-

p(2m )m

(2m

i di!)

where n y• 22 2=li=1 (ndi) and~i m•

(21 )

2mE

licE(Go) didj. i

52

Models of Random Graphs

Proof We shall represent the graphs in 5Y(d; Go) as images of so called configurations. Let W = Uj=I Wj be a fixed set of 2m = • dj labelled vertices, where IWjI = dj. A configuration F is a partition of W into m pairs of vertices, called edges of F. Let 0 be the set of configurations. Clearly

I@1

= N(m) = (2m).2- m = (2m-

1)!!

Given a configuration F e (F, let O(F) be the graph with vertex set V in which ij is an edge iff F has a pair with one element in Wi and the other in Wj. Note that every G e 2Y(d; Go) is of the form G = O(F) for exactly fl-=_ dj! configurations. However, not every O(F) belongs to Y2(d; Go) since Oi(F) may share an edge with Go and the degree of vertex i in O(F) may be smaller than di, either because F contains an edge entirely in Wi or because it contains two edges joining Wi to Wp. In order to identify the set {fr-'(G) : G e Y(d; Go)}, we consider the k-cycles of a configuration. For k E N a k-cycle of a configuration F is a set of k edges, say {ei,e2,...,ek} such that for some k distinct groups Wj, Wj2_..., Wjk the edge ei joins Wji to Wj,+,, where Wjkl - Wj. A 1-cycle is said to be a loop and a 2-cycle is a coupling. With a slight abuse of terminology an edge of F c (F joining classes Wi,Wj with ij e E(Go) is said to be a O-cycle. Then clearly O(F) E =Y/(d; Go) iff Xo(F) = X 1 (F) = X 2(F) = 0. Hence, if we turn (0into a probability space by giving all configurations the same probability, then ndi! + X 2 = O)N(m) Y(d; Go) = P(Xo + X, i=!

Therefore our theorem is equivalent to P(X 0 + X 1 + X 2 = 0) ,- e-/2- 2/4 #.

(2.7)

We shall prove (2.7) by showing that for every fixed k the distribution of (Xo, X 1, ... ,Xk) is close to the distribution of (Yo, Y1,..., Yk), where the Yi's are independent Poisson r.vs with E(Yo) = p and E(Yi) = 2'/2i, i = 1....k. Then Yo + Y1 + Y2 is a P(A1/2+A2/4+ p) r.v., so P(Xo + X 1 + X 2 = 0) - P(Yo + Y1 +

Y2 = 0) =

e-A2-2/4-,

which is exactly (2.7). In order to show that asymptotically the distribution is a Poisson distribution, we look at joint factorial moments. For a ( V put W(O')

=

fdi(di - 1). ica

2.4 Random Regular Graphs Let n 2 denote the number of di's greater than 1, i.e. set n2 n, d, > 2}. Since 2m - n -* oo, we have n 2 -+ 00. Clearly

53 =

max{l I 1<

~Il=k

{£ }k

and a trite calculation gives

. {w(5)

di(di - 1)

/k!

(2.8)

for every k > 1. For k = 1,2.... set 2(k - 1)!

Ck(d)

5

w(O).

(2.9)

oIl=k

Then by (2.8) limn,_.Ck(d)/n k > 0. Furthermore, put _ didj.

Co(d)=

(2.10)

i>j

ijEE(Go)

Clearly there are exactly Ck(d) sets of pairs of vertices that can be k-cycles of configurations. Let us show that if the sequence d is decreased a little then Ck(d) does not decrease too much. More precisely, given a non-negative integer t, denote by d- t the sequence dt+l, dt+2,.... d, and define Ck(d- t) by (2.9) and (2.10). Very crudely indeed, for all k > 1 and 1 t < n we have Ck(d)

-

Ck(d - t) < 2tnk-l A2 kk !

since the edges of a k-cycle join groups of size at least 2. Consequently Ck(d - t)

{1I + o(1)}Ck(d),

(2.11)

{I + o(1)}CO(d)

(2.12)

whenever k and t are fixed. Similarly one can show that C 0 (d-

t)

for every fixed t, provided C0 (d)

-

c, that is e(Go) --. oo.

54

Models of Random Graphs

The probability that a configuration contains a given fixed set of I vertex disjoint (independent) edges is N(m - l)/N(m)

=

1/{(2m - 1)(2m - 3)... (2m - 21 + 1)}.

If I is fixed, this probability is {1 + o(1)}(2m)-'.

(2.13)

Now we can easily calculate the joint factorial moments of the Xi's. Note first that by (2.8), (2.9) and (2.13) we have for every i > 1 Ei = E(Xi) - Ci(d)(2m)-i - i/(2i)

(2.14)

=i

and by (2.10) and (2.13) E0 = E(Xo)

=

Co(d)/(2m - 1)- •

=

o.

(2.15)

If 2o = p = o(1) then P(Xo = 0) = 1 - o(1) and it suffices to consider the joint factorial moments of X 1,X 2..... Hence we may assume that 2o is bounded away from 0. =yr Denote by W(ro, r1, ... ,rk) the set of ordered r-tuples, r = -- =0 ri, whose first r0 elements are 0-cycles, the next r, elements are 1-cycles, etc. Let 16'(ro, rl, ... ,rk) be the subset of 16(ro, rl, ... ,rk) consisting of r-tuples for which no Wi contains endvertices of edges belonging to distinct cycles, and set W6"= 16\1'. Finally, for a configuration F E (D,denote by Y(ro,r 1, ... ,rk)(F) the number of elements of W6in F, and define the r.vs Y' and Y" analogously. Then clearly E{(Xo)ro(Xi)rm

...

(Xk)rk} =

E{Y(ro, rl.... rk)}

(2.16)

and E(Y) = E(Y') + E(Y").

(2.17)

Furthermore, k

k

H Ci(d

t)ri •_ Y'(ro, rl,..., rk)

-

i=0

where t = 2ro + (2.15) we have

H Ci(d)r, i 0

y-

iri. Consequently by (2.11), (2.12), (2.13), (2.14) and k

E{Y'(ro,rli....rk)}

-

{Co(d)ro/(2m)r}

k

I{Ci(d)ri/(Zmyri} i=1

i=0

(2.18) On the other hand, the cycles of an r-tuple counted by Y"(ro, rl,.... rk)

2.4 Random Regular Graphs

55

are such that for some 1 < t = 2r0 +-Z-= 1 iri altogether they have at most 1 edges and these edges join at most 1- 1 groups W,. Hence, very crudely indeed, t

E{Y"(ro, rl,..., rk)J} <

n'-'2At(2m - 2t)-1 = 0(n-1). (=2

Consequently, by (2.16), (2.17) and (2.18) we have k

E (XO)ro(Xi

(Xk)rk

} i=0

Therefore, by Theorem 1.20, Xo,X 1, ... Xk are asymptotically independent Poisson random variables with means 24, Al ... , )-. In particular, P(Xo + X 1 + X 2 = 0) -e0

-2

=

e

as required. Corollary 2.17 Suppose r >_2 is fixed and rn is even. As beibre, denote by Lr(n) the number of labelled r-regulargraphs of order n. Then tr(n)

-

e-(r2_,1)/4 (rn)!/{(rn/Z)!zr,12(r !)n}

• x2e-(r'-1)/4

r•r1

n2

It is not hard to show that A in Theorem 2.16 need not be fixed. The same proof executed with slightly more care shows that the assertion holds for A < (log n)1/ 2/2 (see Bollobfis, 1980b; Bollobfis and McKay, 1985). The main significance of Theorem 2.16 and its proof is that when we study random regular graphs (whose degree does not grow too fast with n), instead of considering the set of regular graphs we are allowed to consider the set of configurations. In order to make this statement a little more precise, we restate some of the facts above for regular graphs. Let dl = d 2 =....= d = r > 2 and let D and 0 be as in the proof of Theorem 2.16. Clearly, 0-

1

(Sr-reg) =

{F E • : X 1 (F) = X 2 (F) = 0}

and O- 1(G)J = (r!)' for every G e 'r-reg-

Given a set (property) Q C Sr-,reg, how can we estimate P(Q)? Find a set Q* ( D such that

Q*n - 1(Sr-reg)

=

O I(Q).

Models of Random Graphs

56 Then, trivially, P(Q)

= =

P(Q*IX 1 =X 2 =o) P(]Q* and Xi = X 2 = 0) 1

1 - P(Q*)

P(Xi = X 2 = O)

P(Xi = X2 = 0)

giving the following simple consequence of the proof of Theorem 2.15. Corollary 2.18 If r > 2 is fixed and Q* holds .br a.e. configuration [i.e. P(Q*) -> 1], then Q holds. br a.e. r-regular graph. Although it does not really belong to this section, let us note another trivial consequence of the proof of Theorem 2.16 (see Bollobfis, 1980b; Wormald, 1981a, b). Corollary 2.19 Let r > 2 and m > 3 be fixed natural numbers and denote by Yj = Yj(G) the number of i-cycles in a graph G c #r-reg. Then Y3,Y 4,. . . . .Yn are asymptotically independent Poisson random variables with means X3 ,24,..., 2 m, where 1i = (r - 1)i/(2i). Proof The assertion is immediate from the fact that X 1, X 2. Xm (random variables on D) are asymptotically independent Poisson random variables. It should be pointed out that L2 (n), the number of 2-regular graphs of order n1 , is considerably easier to handle than Lr(n), r > 3. Here are some results from Comtet (1974, pp. 273-279). Clearly L 2(1) = L 2(2) = 0. A geometric interpretation of L2 (n), due to Whitworth (1901, p. 269), leads to the recursive formula (Robinson, 1951, 1952; Carlitz, 1954, 1960) L 2(n)

=

(n - -)L

2 (n -

1) + (n

1)L

2 (n

-3),

n > 3,

where L 2(0) is defined to be 1. From this one can deduce that the exponential generating function of L2(n) is 1 {rt 2 +±2t} .2 t exp L 2(n)t'/n! =• E 4__ ex 01 nŽO '-

In conclusion, let us say a few works about the considerably more difficult problem of estimating Lr(n) when r = r(n) • n/2 grows with n fairly fast. McKay and Wormald (1990b, 1991) introduced powerful new methods and obtain several remarkable results. As in the case of small r, the problem of enumerating r-regular graphs does not really differ from

Exercises

57

that of graphs with given degree sequences. For d = (di)', write L(d), for the number of graphs on [n] with d(i) = di, i = 1..... n. Thus, if Go is the empty graph then, with the notation of Theorem 2.16, L(d) = IY(d; G0 )J. McKay and Wormald (1990b) gave a complex integral formula for L(D) under the assumption that for a sufficiently small r > 0 and some > we have Idi - dj < n1/2+ and min{di, n -d} > Cn/ log n for all i and j. However, the integral itself is rather difficult to estimate. Using a different approach, McKay and Wormald (1991) gave an asymptotic formula for L(d) under the assumption that A = maxdi = o(m 1 / 3 ), where m = E n di is the number of edges. In particular, they proved the following substantial extension of Corollary 2.17: if I < r = o(n 1/ 2) and rn is even then Lr(fl)

=

exp

(

r 2 -1

r3

4

12n

/

)

(rn)! (rn/2)!2rn/ 2(r!)n"

McKay and Wormald (1990a) also gave an O(r 3 n) algorithm to generate r-regular graphs uniformly when r = o(n1/3).

Exercises In Exercises 1-3 Q is a monotone increasing property. 2.1 Suppose 0 < M 1 < M 2 < N, PM,(Q) < 1 and PM2(Q) > 0. Prove that PM,(Q) < PMA(Q). 2.2 SupposeO
-

Pp(Q)}Pp(Q).

What is Q? 2.3

Let M = [pN] and pqN

-*

oo. Show that

Pp(Q) _ PM(Q)/2 + o(1) and PM(Q) > 2Pp(Q) - 1 + o(1).

2.4 2.5

Prove that both inequalities are best possible. Suppose Q is a convex property, pqN -• oo and almost no Gp has Q. Is it true that almost no graph in W([pN]) has Q? Let H0 be a fixed graph and denote by X(G) the number of subgraphs of G isomorphic to HO. Show that if P1•P2 and M satisfy the conditions in Theorem 2.3, then the conclusion of the theorem holds.

58

Models of Random Graphs [Let V' be the collection of graphs isomorphic to H0 , with vertex set contained in V. For H E * set 1, ifH c G, XH(G 0, otherwise.

2.6 2.7

Note that X = Z-H,,,. XH and Ep,(XH) = Ep,(XH') and EM(XH) = EM(XH,) whenever H,H' e -V. Apply Theorem 2.3 to Epi(XH) and EM(XH).] Imitate the proof of Theorem 2.8 to prove Theorem 2.11. Let Q1 and Q2 be two models of r.gs. Suppose for all 11,c12 .... I I V(2) we have Pl(G c 0 1 contains none of the edges al,02 .... 100 <ý P2(G G 02 contains none of the edges o , a2 .....

2.8

IS).

(Intuitively, an element of 01 is less likely to miss edges than an element of Q2, so an element of K 1 should have more edges than an element of Q 2.) Let Q be a monotone (increasing) property. Does P1(Q) > Pz(Q) hold? Let k > 1 and OC,92 .... , s E V(2),z,i cr1. Show that P(Gk-out does not contain

l)

=

1-

k 1)2

=

q

and P(Gk-out does not contain

2.9

2.10

i, i = 1.

s) < qS.

Consider the following Markov chain (G,)' defined by Capobianco and Frank (1983). The states are finite graphs and the initial state G1 is the trivial graph with one vertex. If Gt- 1 is complete then G, is obtained from Gt- 1 by the addition of a vertex. If Gt-l is not complete then with probability p we add an edge to obtain G, and with probability q = 1 - p we add a vertex. Here 0 < p < I is fixed, and all missing edges are equally likely to be added. [Note that IG, + e(G,) = t for every t.] Show that, with probability tending to 1, Gt has about pt vertices and qt edges. Let r > 3 be fixed and let uo = o(n). Prove that there is a function e = e(n, uo) -- 0 such that if Go = Go(n) is a graph with vertex set V = 1. n}, size u < u0 and maximum degree at most r, then

(1-

)

rn
P(Gr-reg

contains Go) _<(1 - e) (ry.

Exercises 2.11

2.12

59

For a fixed k c N consider Sk-out. Call an edge xy of a random graph a double edge if the original directed graph contains both xcy and yx. Use the technique of the proof of Theorem 2.16 to deduce that the number of double edges tends in distribution to P(k 2/2). Show also that E{e(Gk-out)} = kn - k 2 /2 ± o(1). Let r and s be fixed natural numbers. For mr = ns = N denote by B(m,n;r,s) the number of bipartite graphs with vertex sets U = {x1 .... Xm} and W = {Yj_..y,} such that d(xi) = r and d(yi) = s for all i and j. Show that as N - oo

B(m, n; r, s)

N! N (r!)mn(s!)n

(-)s

11

(B~k~ssy, B~k~ssy and Koml6s, 1972.) [Imitate the proof of Theorem 2.16. Define a configuration as a 1-1 map F of U' = U7ml Ui into W' = U'=I Wj, where lUi = r and Wjj = s. Note that there are N! configurations. For a configuration F denote by X(F) the number of 'double edges', i.e. pairs of elements (a, b) of U' for which a,b G U, and F(a),F(b)G 2.13

Wj for some i and j. Show that X d P{(r - 1)(s - 1)/2}.] For r, s, m, n and N as in Ex. 12, denote by M(m,n;r,s) the number of m x n matrices with row-sums r and column-sums s. Show that as N

-

oo

(m,n;r,s)

2.14

2.15

N! (r )(s (r!)m(s!)n

)2

(B~k~ssy, B~k~ssy and Koml6s, 1972.) [Consider the same configuration space as in Ex. 12 and treat the number of double edges, triple edges, etc., as random variables of the configuration space.] Let r > 0 be fixed. Show that (i) if p = {(2 + r)logn}/n, then a.e. Gp is connected and (ii) if p = {(1 + 8) log n}/n, then a.e. Gp is connected. (The first assertion is quite trivial, the proof of the second needs a little work.) Prove that for every fixed o > 1 and t = [an log nj we have P(G, is connected) = 1 - O(n1-2a).

3 The Degree Sequence

The degree of a vertex is one of the simplest graph invariants. If x C V, then dG(x), as a function of G c N(n,p), has binomial distribution with parameters n - 1 and p. Furthermore, if x,y e V and x =ý y, then dG(x) and dG(y) are rather close to being independent random variables on W(n, p). Thus it is not surprising that one can get rather detailed information about the distribution of individual members of the degree sequence. Erd6s and R~nyi (1959) were the first to study the distribution of the maximum (minimum) degree, and various aspects of the degree sequence were investigated by Ivchenko (1973a), Erd6s and Wilson (1977), Bollob',is (1980a, 1981a, 1982a) and Cornuejols (1981). In the first four sections, based on Bollobis (1981a, 1982a), we present the basic results concerning the degree sequence. The fifth section is devoted to a result of Babai, Erd6s and Selkow (1980) concerning the graph isomorphism problem and to the sketch of a related result of Bollobis (1982b). Throughout the section limf(n) means the limit of f(n) as n --+ oc

through a certain set of natural numbers.

3.1 The Distribution of an Element of the Degree Sequence In this chapter we shall write (di)" for the degree sequence of a graph G c

((n,p) arranged in descending order: d,> d 2 > ... > d,,. Thus

di(G) = A(G) is the maximum degree and d,(G) = 6(G) is the minimum degree. Our aim is to determine the distribution of this sequence. It will turn out that depending on the function p = p(n), different members of this sequence are almost completely determined. In order to study the degree sequence we introduce some related random variables. We write Xk = Xk(G) for the number of vertices of degree k, Yk for the number of 60

3.1 The Distribution of an Element of the Degree Sequence

61

vertices of degree at least k and Zk for the number of vertices of degree at most k. Thus

and

Yk = YX1

Zk = ZX.

l>_k

I<_k

Clearly dr > k iff Yk > r and dn-r < k iff Zk > r + 1.

If p = o(n- 3 /2 ), then n--1c

n-i

E(Y 2)

ZE(Xj) j=

~ pj •

<- n~E

(pn)'/j!

=2

o=)

j=2

so a.e. Gn,p consists of independent edges. Hence if also pn 2 ccd,then a.e. Gnp has degree sequence d, = d2 .... = d2M = 1,d2M+ I ... = d,, = 0, where M = e(Gn,p) "- pn 2 /2. Therefore in the discussion that follows we may assume that neither p = o(n-3 / 2) nor 1 - p = o(n- 3/ 2) holds. Theorem 3.1 Let e > 0 be fixed, En-3/ 2 <_ p = p(n) _ 1 - en-3/2, let k = k(n) be a natural number and set 2 k = 2 k(n) = nb(k;n - 1,p). Then the following assertions hold. (i) If limAk(n) = 0, then limP(Xk = 0) = 1. (ii) IJlim Ak(n) = oo, then limP(Xk > t) = 1 (iii)

for every fixed t. f10 < lim Lk(n) < lim 2 k(n) <

o,

then Xk has asymptotically Poisson distribution with mean P(Xk = r) -,

ýk:

e-4ýr

for every fixed r. Proof We may assume without loss of generality that p < 1/2. Note that 2 ýk = k(n) is exactly the expectation of Xk(G). Hence P(Xk Ž_ 1) •< E(Xk)

=

Ak(n),

so the first assertion follows. From now on we suppose that limruk(n) > 0. The remaining assertions of the theorem will follow from the fact that for every fixed r > 1 the rth factorial moment Er(Xk) of Xk is asymptotic to 2k(n)r.

62

The Degree Sequence

Recall that Er(Xk) is the expected number of ordered r-tuples of vertices (x1, x 2. X xr) such that each vertex xi has degree k. What is the probability that r given vertices xIx 2,...., x, all have degree k? Suppose we have chosen the edges joining the xi's. If there are 1 such edges and vertex xi is joined to di • k vertices xj, 1 j < r, then Zi=1 di -21 and xi has to be joined to k - di vertices outside the set {Xf,X2 .X, Xr}. The probability of this event is

H b(k -di; n -

r, p).

(3.1)

i=1

Let us distinguish two cases according to the size of p(n). Suppose first that p(n) < ½ is bounded away from 0. Then limAk(n) > 0 implies that k(n) is about pn, say 1Ipn 3 <• k(n) < 3pn. [In fact, by Theorem 1.1, we have k(n) = pn + O(nlogn)1 /2 .] From this it follows that for every r E N there is a function Or(n) such that limn-- 0,r(n) = 0 and b(k-d;n-r,p) -1 t (n). b(k;n-

1, p)

Therefore, by (1), E,(Xk) - (n)rb(k;n - 1,p)'

-

.ý.

(3.2)

Now suppose that p = o(1). Then (1) implies the following upper bound for E,(Xk): Er(Xk) _(n

/=0~

plqR-q

i ~max r,•d,* -21 ]Jb(k-d*;n-r,p),

(3.3)

where R = (2) and the maximum is over all sequences dl, d;,..., d with I d* = 21 and 0 < d* < min{r - 1,k}. Clearly lim 2k(n) > 0 implies that k(n) = o(n), so if 0 < d < min{r - 1,k} and n is sufficiently large, b(k-d;n-r,p)< b(k;n--r,p)

(k)d

- (n--

p-d <2(k)d

k)d

pn

"

Therefore (3) gives Er(Xk)

•

nrb(k;n-r,p)r {1 +

()

p 12r (k

)2,}

3.1 The Distributionof an Element of the Degree Sequence < nrb(k;n- r,p)r{

63

}

+2r2•-

(3.4)

Our assumption that lirn k(n) > 0 implies that k2 o(pn2 ). Indeed, if for some fixed q > 0 the inequality k >_ p 1p/2n held for arbitrarily large n then we would have -

4iM k (n)

_ limn n(.) pk < limn (efn)kpk 'IIlk/ n( e lk q1n p12 n kepnnk <-lim =lim n < lim n 2-n"' = 0,

contrary to our assumption. As k 2 = o(pn 2 ), inequality (4) implies Er(Xk) <_ýnrb(k; n - r, p)r{1 + o(1)} =- A,•{ I÷

o(1)1.

By considering only independent r-tuples of vertices of degree k we find that Er(Xk) > qR(n)rb(k;n - r,p)r = r{1 + o(1)}, since q

1, sso

(3.4')

Er(Xk) "-• A'.

Thus the rth factorial moment of Xk is asymptotic to the rth factorial moment of P(4). Now if lim k(n) = o, then E2(Xk) - Ak implies that E(X 2 ) =-A2{1 + o(1)} so the Chebyshev inequality gives lim P(Xk _ t) = 1 n--0O0

for every fixed t. Finally, if lirM k < oc, then by Theorem 1.20 relations (3.2) and (3.4') imply that asymptotically Xk has Poisson distribution with mean 2 k. LI As an immediate consequence of Theorem 3.1 we see that if almost every G,p has a vertex of degree k, then for every fixed t almost every Gn,p has at least t vertices of degree k. The proof of Theorem 3.1 can be modified to give analogous results about Yk and Zk. Theorem 3.2 Let e > 0 be fixed, en-3/ 2 < p = p(n) <_ 1 --nk = k(n) be a natural number and set k = nB(k;n- 1,p) and Vk = n{1 - B(k + 1;n-

1,p)},

312

, let

64

The Degree Sequence

where B(l;m,p)

=

Zb(j;m,p). j>1

Then the assertions of Theorem 3.1 hold if we replace Xk and and 14k or Zk and Vk.

uk

by

Yk

D]

This last result enables us to determine the distribution of the first few and the last few values in the degree sequence. First we present some results concerning the case when p is neither too large nor too small. Theorem 3.3 Suppose p = p(n) is such that pqn/(logn)3 -* oc. Let c be a fixed positive constant, m a fixed natural number and let x = x(n, c) be defined by n eX2/2 = c. (3.5)

,2x

Then m-1 ck

lim P{d. < pn + x(pqn)1/ 2} = e-c E

c

k=O

Proof Put K = [pn + x(pqn) 1 21]. Then dn < pn + x(pqn)1/ 2 means that at most m - 1 vertices have degree at least K, that is YK

m - -1.

By Theorem 3.2 the distribution of YK tends to P(PK), where ,UK

= nB(K; n - 1, p).

Now by the definition of x we have x(pqn)1 / 2 = o{(pqn) 21 3}, so by Theorem 1.6 ,UK

- (c/n)n = c.

This implies the assertion of the theorem.

D

For every fixed positive c the value of x calculated from (5) is about (2 log n) 1 /2. Furthermore, x(n, 1) = (2 log n)1/

2

1

log log n I. 4logn

log(27rl/ 2 )

2logn

+° (g

f

n)-1/2} 112.

3.2 Almost Determined Degrees

65

Hence x' = x(n, 1) + 6 corresponds to c' - exp{-(2 log n) 1/26 }, provided 6 = o(1). Therefore Theorem 3.3 can be rewritten in the following form. Theorem 3.3' Suppose pqn/(log n) 3 -+ x and y is a fixed real number. Then for every fixed m we have n 2 1- loglog n limp dm
+ Y -1g2) +1g log(27r'/

4logn

2logn

2logn

e e ' Ee-'Y/k!.

El

k=O

Corollary 3.4 Suppose pqn/(logn)3 - oc and y is a fixed real number. Then the maximal degree A = d, of Gn,p satisfies limp A
- 1/2) +• 21g 4olog n _ Iog(2ir log 4logn

n-

2logn

y

2logn

If p is close to 0 (or 1), then we rely on Theorem 3.2 for information on the maximum and minimum degrees. In the most interesting case both these degrees are bounded. The proof is left as an exercise (Exs. 1 and 2). Theorem 3.5 Let k, x and y be fixed, k > 2, x > 0 and y G IR. If p - xn-(k+l)/k then P{A(Gp)

=

k}

*

1

-_

eXk lk/I and P{A(Gp) =k--1}-,e

-x/k!.

if p = (log n + k log log n + y)/n

then P{6(Gp) =k} -1->I

ee /k! and P{6(Gp) =k+ ±1}+-

e -e /k!.

l

3.2 Almost Determined Degrees In what range of p = p(n) is it true that almost all graphs G E S(n,p) have the same maximum degree? What is the corresponding range of p if almost all GP have the same minimum degree? In what range of p is

66

The Degree Sequence

it true that a.e. Gp has a unique vertex of maximum degree? The aim of this section is to answer these questions. The proofs of the results are based on Theorem 3.1. We show first that if almost all graphs have the same maximum degree then p = o(log n/n) and then we show that this necessary condition is essentially sufficient. Theorem 3.6 Suppose that p <_ 2 and there is a Junction D(n) such that in the space W(n,p) we have lim P{A(G) = D(n)} = 1.

n-- c,

Then p = o(log n/n). Proof By Theorem 3.1 we must have lim limn-, nb(D + 1);n - 1,p) = 0. Therefore

,nnb(D;n - 1,p) = oo and

lim b(D + 1; n - 1, p)/b(D; n - 1, p) = lim p(n - D -1) .... o q(D + 1)

0,

so pn/D -- 0. Using again that lim nb(D;n - 1,p) = oo, we find that for every c > 1 lim nb([cpn] ;n- 1,p) = oo

n-'om

and so lim n(epn/cpn)cPn = lim n(e/c)cpn = cc. l---oo

n-o

As this holds for every c > 1, we have p = o(log n/n). Theorem 3.7 Suppose p = o(log n/n). Let k = k(n) > pn be such that max{2k(n), uk(n)- 1 } is minimal. Then the following assertions hold. (i) Iff0 < lim)k(n) < lim nk(n) < c, then P{A(G) = k(n)}

II - e-;k(n)

and

P{A(G) = k(n) - I}

e -k(n).

(ii) If lim k(n) = cc, then a.e. GP has maximum degree k(n) and if lim lk(n) = 0, then a.e. Gp has maximum degree k(n) - 1. Proof It is easily checked that k/pn --* o, 4k+14 -/ pn/(k + 1) 4k/kl -•pn/k

--

0. From these it follows that 4k+1

-.

2

--

0 and k1

0 and -*

c0.

3.2 Almost Determined Degrees

67

By Theorem 3.1 the relation 2 k-1 - c implies that a.e. GP contains a vertex of degree k(n) - 1 and so the maximum degree of a.e. Gp is at least k(n) - 1. Furthermore

Sn--1 !

P{A(G) > k +I}

=

P (E

XJ > 1

n-1

E

E(X j )

=

O(k+±)

=

o(1).

j~k+l

( j=k+l

Hence a.e. Gp is such that the maximum degree is k(n) - 1 unless there is a vertex of degree k(n), in which case it is exactly k(n):

P{A(G) = k(n)} ,-, P(Xk

> 1)

and

P{A(G)

=

k(n) - 1}

1 - P{A(G)

=

k(n)}.

Finally, by Theorem 3.1, we know the behaviour of P(Xk >_ 1).

[]

If k0 is the maximal integer with 2k0 > 1 then k is k 0 or k0 + 1, so the maximum degree of a.e. Gp is k0 - 1, ko or k0 + 1. It is easily seen that if p = o(log n/n) and pn3/ 2 -> c•, then p can be altered slightly so that almost all graphs have the same maximum degree. To be precise, if O(n) = o(log n/n) and O(n)n3/ 2 --. oc, then one can choose functions p = p(n) - O(n) and k(n) such that a.e. GP has maximum degree k(n). In what range of p is the maximum degree of a.e. Gp a fixed natural number k? The results above show that this range is exactly {p " 0 < p < 1,pn(k+ 2 )/(k+l) -+ 0 and pn(k+l)/k _ o} If p _<2, then the minimum degree behaves somewhat differently from the maximum degree. We have the following analogue of Theorems 3.6 and 3.7. Theorem 3.8 (i) Suppose that p •_ in the space S(n, p) we have

2

and there is a function d(n) such that

lim P{6(n) = d(n)} = 1. Then p • {1 +o(1)}logn/n. (ii) If p {1I + o(1)} log n/n, then there is a function d(n) such that the minimum degree of a.e. Gp is either d(n) or d(n) - 1.

The Degree Sequence

68

Proof (i) Write p in the form p = {log n + wo(n)}/n. Then, if ow(n) O(logn), say,

=

AO= nb(O; n - 1,p) = n(1 - p),-1 - nn- 1 e-"(') = eso, by Theorem 3.1, Gp almost surely has an isolated vertex iff wo(n) -oo. Therefore we may assume that p > (log n + C)/n for some constant C. As in the proof of Theorem 3.6, we find that if limn_.

P {16(n)

=

d(n)} = 1, then for every fixed P, 0 < E < 1, we have n(Epn)-y/

2

(e/ep) :P"p•pP

e pn+p2Hn _

00.

Hence

As this holds for arbitrarily small positive values of e, it follows that p _
then a.e. Gp has a unique

vertex of maximum degree and a unique vertex of minimum degree. (ii) If p < 1 and a.e. Gp has a unique vertex of maximum degree or a unique vertex of minimum degree, then pn/ log n ---. oc. Proof (i) For simplicity write b(l) = b(1; n - 1, p) and B(1) = B(1; n - 1,p).

Let k be the maximal natural number with nB(k) > 1. It is easily checked that k > pn, k - pn, nb(k-1) --4 0 and b(k)/b(k-1) 1. Consequently one can choose 1 = 1(n), pn < 1 < k, such that nb(l) -- 0 and nB(l) -* oo. If we vary I in such a way that these relations remain valid then we can find an m = m(n), pn < m < k, such that nB(m) --+ oo and nB(m)nb(m) ---* 0.

We claim that a.e. Gp has a vertex of degree at least m, but almost no Gp has two vertices whose degrees are equal and at least m. Indeed, the expected number of vertices of degree at least m is E(Ym) = nB(m). Hence E(Ym) --* oo and by Theorem 3.2 we have A(G,) > m for a.e. Gp.

3.3 The Shape of the Degree Sequence

69

On the other hand, the probability that there are at least two vertices whose degrees are equal and at least m is at most E P(Xj > 2) j>m

_
2 n 2b(j;n- 2,P)

E 2 (Xj) <_ 1

j!m

j>m-1

< nb(m - 1; n - 2,p)nB(m - 1; n - 2,p) ,-. nb(m)nB(m).

To see the second inequality above, note that the expected number of ordered pairs of vertices of degree j _>m is n(n - 1){pb(j - 1; n - 2, p) 2 + (1 - p)b(j; n - 2,p) 2} • n 2 b(j; n - 2, p) 2 . Hence

Zj>_m

P(Xj > 2)

--

0, proving the assertion.

(ii) A sharper form of this assertion is left as an exercise for the reader [Ex. 3, see also Bollobhis (1982a)]. El Let us summarize the results of this section in a somewhat imprecise form. If p = o(log n/n), then both the minimum and maximum degrees are essentially determined and there are many vertices of minimum degree and many vertices of maximum degree. If elog n/n < p < {1 + o(1)} log n/n for some e > 0, then the minimum degree is essentially determined and there are many vertices of minimum degree, while the maximum degree is not confined to any finite set with probability tending to 1. If(1 +e)logn/n <_p < Clogn/n for some e > 0 and C > 0, then neither the minimum nor the maximum degree is essentially determined and the probability that there are several vertices of minimum and maximum degrees is bounded away from 0. Finally, if pn/ log n - oo and p<½,then a.e. Gp has a unique vertex of maximum degree and a unique vertex of minimum degree.

3.3 The Shape of the Degree Sequence Having investigated the smallest and largest members of the degree sequence, we turn to the study of a general member dm of the degree sequence. Recall that Yk = jIk Xi is the number of vertices of degree at least k in Gp. The mean of Yk is Yk = nB(k;n - 1,p). We shall use Chebyshev's inequality to show that in a certain range of p and k most graphs are such that Yk is close to Pk. Our main result is based on two lemmas. For simplicity we shall omit p from b(I;N,p) and B(l; N,p). Lemma 3.10 The variance Of Yk satisfies a 2 (yk) = E{(Yk -- Pk) 2} <:

Ik

+-n2pqb(k + - 1;n- 2)2.

70

The Degree Sequence

Proof' The second factorial moment of Yk is easily calculated: E 2 (Yk) = n(n - 1){pB(k - 1; n - 2)2 + qB(k; n - 2)2}.

(3.6)

Indeed, the probability that both of two given vertices of Gp, say a and b, have degree at least k is

P{ab c E(Gp) and dGp-b(a) > k - 1,dGp-a(b) Ž_ k - 1} + P{ab E(Gp) and dG,,4(a) Ž_ k, dG,,-(b) > k}. The events {ab e E(Gp)}, {dG -h(a) >_ k - I} and {dG• a(b) >_k - 1} are independent, so the probability of their intersection is pB(k-1;n-2)2 .

P{ab E E(Gp)}P{dGp-b(a) > k-1}P{dGp-a(b) Ž_ k-1}

The second term in the expression (3.6) for E2((Yk) is justified analogously. Hence o0(Y)

E(Yk2) - k lk + n(n-- 1){pB(k-

1;n- 2)2 + qB(k;n- 2)2} -/Pk < P1 k + n2 {pB(k - 1;n - 2)2 + qB(k;n - 2)2 - B(k;n=

1)2}. (3.7)

Note now that B(k;n-- 1) = pB(k-- 1;n-

2) + qB(k;n-

2)

and B(k-

1;n-2)-B(k;n-2) =

(-

2

p

1

qn

1

=b(k-

1;n-2).

Substituting these into (3.7) we find that c"2 (Yk)
+ n 2pqb(k - 1;n - 2)2.

Lemma 3.11

U-2 (Yk)

= O(/uk).

Proof This is immediate from Lemma 3.10, since one can check from Theorems 1.1-1.5 that n 2pqb(k - 1; n - 2) 2 = O{nB(k; n - 1)}. Theorem 3.12 Suppose m = m(n) = o(n), re(n) --+ co and wo(n) -> oo arbitrarily slowly. Define x by I - (D(x) = m/n.

3.3 The Shape of the Degree Sequence

71

Then a.e. Gp satisfies Id. - pn

-

1 12

x (pqn)

Proof The assumptions imply that x have m , _1

n

-

n

ml

] < w(n)

1/2

oo, so by (15') of Chapter 3 we e X212 -

2jrx

and x - {2 log(n/m)} 1

2

.

We may assume without loss of generality that co(n) < mI/ 2 and ow(n) = o(x). Set k = pn + x(pqn)1 /2 + )(n){pqn/mlog(n/m)}j1 2 . By Theorem 1.6 E(Yk) - mexp[-xeo(n)/{mlog(n/m)} 1/ 2 - wo(n) 2/2mlog(n/m)]

-*

o

and m - E(Yk)

'-

m(1 - exp[-xw)(n)/{m log(n/m)}1 / 2 - wo(n) 2 /2m log(n/m)])

- mxow(n)/{m log(n/m) }11 / 2 >

w)(n)m 1/2 .

Hence m >_ E(Yk) + wo(n)E(Yk)"

2

.

By Lemma 3.11 and Chebyshev's inequality we obtain that Yk < m in a.e. Gp, so a.e. graph Gp satisfies dm< k = pn + x(pqn)1 /2 + wo(n){pqn/m log(n/m)}I

2

.

A similar argument gives that for a.e. Gp we have d, > pn + x(pqn) 1 2 - o(n){pqn/m log(n/m)}1

2

.

If m does not grow too fast, then the dependence of x above on m can be calculated precisely and one is led to the following corollary.

72

The Degree Sequence

Corollary 3.13 Let wo(n) --+ oo and m = m(n) = O(1og n)/(log log n) 4 . Then a.e. Gp satisfies n)1/ 2 + log log n (,n dm -pn - (2pqn log

1/2

( 8log )nJ 12

+ log(2r / m)

pqn

(on

1/2

G2logn)

•

•

pqn

1/2

(n mlogn)

Another assertion implied by Theorem 3.12 is that if p is not too small, then almost all Gp's have about the same maximal degree. Corollary 3.14 If pn/ log n -* co, then a.e. Gp satisfies A(Gp) = {1 + o(1)}pn. Proof Taking m = 1 and wo(n) = (pn/logn)1/2 in Theorem 3.12, we see that x _<(2 log n) 1 / 2 and so 1/ 2 2 IA(Gp) - pnl< {2(log n)pqn}"/ + wo(n)(pqn/ log n) 1/ < O(pn) + to(n)(pn/ log n)

2

almost surely. The result above is best possible (Ex. 8). Furthermore, if (pn log n)/ log log n -* x, then almost all Gp's have about the same minimal degree (Ex. 9). McKay and Wormald (1997) proved several powerful general results about the joint distribution of the degrees of a random graph that imply the results above. They showed that the joint distribution can be closely approximated by models derived from a set of independent binomial distributions. For example, they proved that for a wide range of functions M = M(n), the distribution of the degree sequence of a random graph GM e W(n, M) is well approximated by the conditional distribution {(X1..... X,) : IZX 1 = 2M}, where X 1, ... ,X, are independent r.vs, each with the same distribution as the degree of one vertex:

k( M )(5k

3.4 Jumps and Repeated Values We have seen that if p is not too close to 0 or 1, then a.e. GP has a unique vertex of maximum degree. In fact, unless p is close to 0 or 1, for a.e. Gp

3.4 Jumps and Repeated Values

73

the first several values of the degree sequence are strictly decreasing. The following result gives a good estimate of the likely jumps di - di+,. Theorem 3.15 Suppose m = o(pqn/logn)1/ 4 , m --* oo and o(n) a.e. Gp is such that di -di

logn)/

_ i•n)

00. Then

for every i < m.

Proof As in the proof of Theorem 3.12, let x be defined by

1 - (1)(x) = m/n. 1 2

Note that x = O{logn) / }. By Theorem 3.12, a.e. Gp satisfies dm Ž: pn + (x - g)(pqn) where r = {log(n/m)}l 1 2. Therefore the theorem will be proved if we show that almost no Gp contains an ordered pair of vertices (u, v) such that K = [pn + (x - e)(pqn)1/ 2 ] <_ d(u) < d(v) < d(u) + J - 1, where

-

oc(n) pqn 1/j2 i 2 G ogn J

The expected number of such ordered pairs is clearly Ln-2 L

b(1;n-2) b(k;n-2)k+J-E

p Z

=n(n -1)

k=K --1

n-2

I=k

k+J-1

b(k;n-2) E

+q k=K

b(l;n-2)

l=k

n-2

- n2

5

b(k; n - 2)Jb(k; n - 2)

k=K--1

• n2 JB(K

-

1;n-

2)b(K

-

1;n--2).

By Theorems 3.2 and 6 of Chapter 3 this gives that if n is sufficiently large, then L i(n) n2 -n 2

pqn logn)

/2

x

m_

(pqn)'/ 2 n

2c(n)x (log n) 1/ 2

Since oc(n) -+ 0, the expected number of pairs tends to 0.

El

74

The Degree Sequence Let us state without proof that Theorem 2.15 is essentially best possible.

Theorem 3.16 Suppose m --* oc and o)(n)

di - di~

<

Sw(n) 2 im

(pqn \logn)

-*

oc. Then a.e. GP is such that

1/2

for some i < m.

3.5 Fast Algorithms for the Graph Isomorphism Problem How fast an algorithm can one give in order to decide whether two given graphs are isomorphic? It is not known that there is such an algorithm whose running time is bounded by a polynomial of the number of vertices (or edges). However, Babai, Erd6s and Selkow (1980) gave a very simple algorithm which, for almost every graph G of order n, tests for every graph H in time 0(n 2 ) whether G is isomorphic to H. This algorithm is based on the fact that the degree sequence of a.e. graph contains many gaps (cf. Theorem 3.15). Let * be a set of graphs with vertex set V = {1,2_..n}. A canonical labelling algorithm of .% is an algorithm assigning to each G G -*a permutation 7TGof V such that G and H in JV' are isomorphic if and only if 7EG(G) and 7UH(H) coincide. In other words, a canonical labelling algorithm relabels the vertices in such a way that two graphs are isomorphic if and only if the new labelled graphs coincide. The next result is a slight extension of the result of Babai, Erd6s and Selkow (1980); it is an easy consequence of Theorem 3.1. Theorem 3.17 Suppose 0 < p = p(n) < 1 is such that p 5 n/(log n) 5

cc. 0

2

There is an algorithm which, for almost every graph GP, tests in 0(pn 2) time Jor any graph H whether H is isomorphic to Gp or not. Proof First we give an algorithm which constructs a set c 65(n, 1 p) such that P(.•) -+ 1. Then we construct a canonical labelling algorithm for MX'. Both of these algorithms will work in 0(pn2) time. Set m = [3log(ll/q)n]. Compute the degrees of the vertices of Gp and then order the vertices by degree: x 1 , x 2, ... x, are such that d(xi) = dl > d(x 2 ) = d 2 Ž! ... Ž_ d(xn) = dn. If di < di+1 + 2 for some i :< m, then put Gp into <S(n, p) - -t and end the algorithm. Otherwise for i = 1, 2,..., n compute ,

f(xi)

=

E a(i,j)2j, j=1

Exercises

75

where a(i,j) is 1 if xixj E E(Gp) or i = j and 0 otherwise. If f(xi) = f(xj) for some pair i, j, 1 _•i < j _•n, then put Gp into <(n, p) - A. Otherwise put Gp into A-.

It is clear that this algorithm runs in 0(pn2) time. Furthermore, straight-forward calculations show that the conditions of Theorem 3.15 are satisfied with c(n) = 3m2(log n/pqn)1/ 2. Consequently, by Theorem 3.15 almost no graph is discarded because di •_ di+l + 2 for some i < m. What is the probability of discarding a graph Gp because of f(xi) = f(xj)? If f(xi) = f(xj) then xi and xj are joined to exactly the same vertices in Vij = {xl,X 2 , .. ,Xm} - {xi, xj}. Let Wij be the set of m - 2 vertices of highest degree in Gp - {xi, xj}. Then Wij ( V11 since di > d2 + 3 > d2 > d3 + 3 >_... >_dm >_dml + 3. Since Wij is independent of the edges incident with xi and x1, P{f(xi) = f(xj)} <_ P(xi and xj are joined to the same vertices in Wi,)

_•qm-

2

= 0(n- 3).

Consequently, the probability of discarding a graph the second time round is not more than (2) q- 2 = 0(n-'). Hence P(A-) --* 1 as claimed. The required fast canonical labelling algorithm is easily constructed. Let xi(,),

X.(

2 ),

... , x,(n) be the order of the vertices according to their

f-values: f(x,(l)) > f(x(2)) > ... > f(x(n)) and label x,(i) by m(i). Clearly two graphs in A` are isomorphic iff the new labelled graphs coincide. El In the theorem above we did not care about 1 - P(3)f, the probability of rejection. All we wanted to ensure was that 1- P(-) = o(1). Inspired by the above theorem of Babai, Erd6s and Selkow (1980) a number of fast algorithms were obtained by Lipton (1978), Karp (1979) and Babai and Ku~era (1979) which have exponentially small probability of rejection. A canonical labelling given by Babai and Ku~era (1979) runs in 0(n 2) time, with probability of rejection cn for some fixed c > 1. Furthermore, the rejected graphs are handled in such a way as to give a canonical labelling algorithm of all graphs with expected time 0(n 2). Exercises 3.1

Let x be a positive constant, k > 2 a fixed integer and p p(n)

-

=

xn--I/k. Then a.e. Gp has no vertices of degree k + 1

and the number of vertices of degree k tends to PA, where 2 = xk/k !. Deduce that if for fixed k we have pn1+1/k -- oo and pn(k+2)/(k+i) - 0, then a.e. Gp has maximum degree k.

76

The Degree Sequence

3.2

Suppose y c R, k Ž> 0 and p = {log n + k loglog n + y + o(l)}/n. Then a.e. Gp has minimum degree at least k and the number of vertices of degree k tends to P,, where p = e'Y/k! In particular, P{6 (Gp) = k} -1 1- e-/ and P{1(Gp) = k + 1} -* e-. Deduce that a.e. Gp has minimum degree at least k + 1 iff p = {log n + k log log n + wo(n)}/n where wo(n)

3.3

->

oo.

Let 0 < c < oo be fixed and p = {c + o(1)}logn/n. Let ko = (1 + tl)pn be the maximal integer with 2o = nb(ko;n - 1,p) _ 1. Set kj = k0 + j and write Xj = Xj(Gp) for the number of vertices of degree ki. Then any fixed number of the Xj are asymptotically independent Poisson random variables with means 2j = 2o(1 + q)J, i.e. whenever ji < j2 < ... < jt are fixed integers and rl, r2 , rt are fixed non-negative integers, (X=i ri, 1 < i < t) --+ e-. fJ-i/ri!, where ,l

3.4

=

Z

)ji. ý=I (Bollobts, 1982a.)

Let c and p be as in the previous exercise. Let '1A unique positive root of 1 +ctl

=

=

qA(c) be the

c(1 -+ q/)log(1 +±q)

and, for c > 1, let tq6 = t16(c) be the unique root satisfying -1 < q6 < 0 (see Figure 3.1). Show that a.e. Gp is such that A(Gp)

-

(1 + fA)clogn and 6(Gp) -(1(I+

q)clog n.

1.0

0.5

0.0

-0.5-

-1.0 0

10

30

20

Fig. 3.1. The functions

iqA(c)

and q6(c).

40

Exercises 3.5

Let c > 0 and p = p(n) - c/n. Prove that for a.e. Gp the maximum degree satisfies AG-lon logn

11+ I log loglog n +. 1 + logc + o(l)f

log log n 3.6

log log n

Let k e N. Show that a.e. maximum degree logn 1+ log logn

3.7

77

Gk-out

log logn

has minimum degree k and

logloglogn+ log log n

1 +logk+o(1) log log n

Write Bp and BM for random graphs from ý{K(n,n),p} and '{K(n,n),M}. Show that if x G R, p(n) = {logn + kloglogn + x + o(1)}/n and M(n) = (n/2){log n +k log log n + x + o(1)}, then lim P{6(Gp) = k} = lim P{1(GM) = k} = 1 --

fl--c

2C-x/k!

n---oc

and lim P{6(Gp) = k + 1} = n---eo lim P{5(GM) = k + 1} = e 2e'/k.

n--oo

3.8 3.9

3.10

Prove that Corollary 3.14 is best possible: if almost all G 's have about the same maximal degree, then pn/ log n --+ oo. Show that if (pn - log n)/ log log n --+ oo, then there is a function d(n) -* o such that 6(Gp) = {1 + o(1)}d(n) almost surely. [Note that d(n) need not be pn.] Let VI, V2 , ... , Vt be disjoint copies of V = {1,2,. .. ,.n}. Select Vk(2) at random and join i to j by as many elements of edges as the number of copies of ij have been selected. Denote by Gtn the random multigraph obtained in this way and let Gs,t,N be the graph in which i is joined to j iff in Gt,N they are joined by at least s edges. Use the connection between GN and Gp to prove that if N = n 11 st [{logn + c + o(1)}/ s then the distribution of the number of isolated vertices of G,,t,N tends to P(e-e). (Godehardt, 1980, 1981.)

4 Small Subgraphs

Given a fixed graph F, what is the probability that a r.g. Gp (or GM) contains F? The present chapter is devoted to a thorough examination of this question. The degree sequence of a graph G e I" depends only on the number j < n,z j = i} c V( 2 ), i = 1 ... n. of edges in the sets Vi( 2 ) = {ij : 1 Our job in the previous chapter was made fairly easy by the fact that these subsets VJ(2) are not very numerous and not very large (there are n of them, each of size n - 1) and that they are essentially disjoint (any two of them have one element in common). The property of containing a fixed graph F has similar advantages: it depends on the edge distribution in fairly few, fairly small and essentially independent subsets of y(2).

Erd6s and R~nyi (1959, 1960, 1961a) were the first to study the appearance of small subgraphs; further results were obtained by Schfirger (1979b), Bollob~is (1981d), Palka (1982) and Barbour (1982). In the first section we shall study a rather special class of subgraphs. As a corollary of the results, in the second section we shall determine the threshold function for the existence of an arbitrary subgraph F. It will turn out that the threshold function depends only on the maximum of the average degrees of the subgraphs of F. The third section is devoted to the case when a r.g. is likely to contain many copies of our graph F. The first two sections are based on Bollobfis (1981d), the third section contains results from Barbour (1982). The study of the number of subgraphs in a r.g. generated by some underlying geometric structure is related to the topic of this chapter but is outside our scope (see Hafner, 1972a,b; Schtirger, 1976). 78

4.1 Strictly Balanced Graphs

79

Fig. 4.1. F1 is balanced, F 2 is strictly balanced and F3 is not balanced. 4.1 Strictly Balanced Graphs The average degree or simply degree of a graph F of order k and size 1 is d(F) = 21/k. The maximum average degree of a graph H is defined as m(H) = max{d(F) :F c H}. A graph H is balanced if m(H) = d(H) and it is strictly balanced if F ( H and d(F) = m(H) imply that F = H. Note that trees, cycles and complete graphs are strictly balanced. It is fairly easy to determine the asymptotic distribution of the number of subgraphs isomorphic to a strictly balanced graph. The results below, from Bolloblis (1981d), extend some theorems of Erd6s and R~nyi (1960). For simplicity we call a graph F an H-graph if F - H. Similarly, F is an H-subgraph of G if F ( G and F • H. Theorem 4.1 Suppose H is a fixed strictly balanced graph with k vertices and I > 2 edges, and its automorphism group has a elements. Let c be a positive constant and set p = cn-kIl. For G e S(p) denote by X = X(G) the number of H-subgraphs of G. Then X we have

P(c'/a), that is, for r = 0, 1,...

d

lim Pp(X = r) = e-,Ar/r!, n-ooI

where 2 = cl /a. Proof Note first that

(X) ..

. ka

a

%. a

(4.1)

80

Small Subgraphs

Indeed, there are (Q) ways of selecting the k vertices of an H-graph from V = {1,2, ..,n} and then there are k!/a ways of choosing the edges of an H-graph with a given vertex set. The probability of selecting 1 given edges is pl. By Theorem 1.20 it suffices to show that for every fixed r, r = 0,1,..., the rth factorial moment Er(X) of X tends to 2 r as n -> oc. Now let us estimate Er(X), the expected number of ordered r-tuples of H-graphs. Denote by E'(X) the expected number of ordered r-tuples of H-graphs having disjoint vertex sets V1, V2,..., Vr. Then, clearly, E'(X)= (n) (n-k) .. (n( -(k-1)k) (kH) r= (n)rkprPH _r. (4.2) Therefore all we need is that E7(X) = Er(X) - E'(X) = o(1). This will follow from the fact that H is strictly balanced. Suppose the graphs A and B are such that B is an H-graph and it has exactly t vertices not belonging to A. If 0 < t < k, then e(A U B) > e(A) + tl/k + 1/k.

(4.3)

Indeed, since H is strictly balanced, fewer than (k - t)l/k edges of B join its k - t vertices in V(A) n V(B). Let H 1, H 2, ... , Hr be H-graphs with IU=rl IrEr V(Hi)l = s < rk. Suppose Hj has tj vertices not belonging to UJiJ• Hi, so that 2 = s - k. We assume that 0 < t 2 < k, so by (4.3) e(H1 u/-H2) I + t• + I. k k~ Since also e (YiHi)

_e

U Hi) +tjl/k,

we have (r\

e

UH)

Si >

1

k- -

.

Consequently, counting rather crudely, for r > 2 we have El( )

kr-1 (n

s

! r

,klk

s~k

kr-1

kr-i

E O(nspsI/klh/k) =l s=k

O(n- 1/I).

(4.4)

s=k

In particular, E'(X) = o(1), so the result follows.

I

4.1 Strictly Balanced Graphs

81

Because of Theorem 2.2, the result above implies the corresponding

assertion about the model ý{n,M(n)}. Corollary 4.2 Let H, k, 1, a, c and X be as in Theorem 4.1 and let M(n) (c/2)n2-k/l. Then lim PM(X = r) = e-;r/r! nf---oo

for every r, r = 0, 1 .... Proof Let 0 < cl < c < c2 and pi = cin-k/l, i whenever G c G', Ppl(X Ž_ r) • PM(X -_ r)

-

1, 2. Since X(G) •_ X(G')

{1t + o(1)}PP2 (X >_ r).

Hence the result follows from Theorem 4.1 if we recall that cl and arbitrary numbers satisfying 0 < cl < c < c 2.

c2

are

The conditions in Theorem 4.1 can be weakened to include the case when H depends on n but k and I grow rather slowly with n. Theorem 4.3 The conclusions of Theorem 4.1 hold if H, k, 1, a and c depend on n, provided

c'/a

"'

for some positive constant A and for every r c N we have krkl = o(n).

Proof All we have to check is that E' - 2r and El' = o(1) continue to hold. The first is immediate: =

k!)rri

(n) (n-k) ... (n-(r-1)k)

so (cI/a)r > E' > (cI/a)r

rk)rk > (cI/a)r

To see the second, recall that

Ell

rk-_. 1

s=k+l

- r

=

l {(s) }!ýrplk QkI() k

rk--1

<

E

(s!/a)rcn-l/I <- c(rk)r'3kn-ll.

s=k+l

By the conditions this is o(1).

{o(1)}(c/a).

82

Small Subgraphs

It is worth noting that although c = c(n) is no longer a constant, it is not far from being one, so the order of p is still about n-k/l and the order of M is about n2-k/1. Indeed, if 2 is a constant and c(n) = (2a)l/1 ,

k !, for large k we have then since 1 _
k11

The main condition in Theorem 3 is satisfied if k grows rather slowly with n. Corollary 4.4 The conclusions of Theorem 4.1 and Corollary 4.2 hold if H, k, 1, a and c depend on n, provided c'/a and

-'.

2.for some positive constant 2

k = O{(logn)1 / 4 }.

Although the results above concern subgraphs which need be neither induced nor disjoint, in fact the subgraphs we get are almost surely disjoint induced subgraphs. Corollary 4.5 Let H, k, 1, a, c and 2 be as in Theorem 4.3 and denote by Xd = Xd(G) the number of vertex disjoint induced H-graphs of G. Then for p = cn-k/l and r = 0, 1,... we have lim Pp(Xd - r) = e-2r/r!.

Proof Denote by Y = Y(G) the number of subgraphs of G having at most 2k vertices and degree greater than 21/k. Since there are only finitely many such graphs, Ep(Y) = o(1).

Consequently a.e. G E S(p) that contains exactly rH-graphs is such that El its H-graphs are induced and vertex disjoint. If H is a tree, then the subgraphs isomorphic to H turn out to be components of GP. 3, let a be the order of the Corollary 4.6 (i) Let H be a tree of order k >>_ automorphism group of H and let c > 0. Then for p = cn-k/(k- 1) a.e. Gp is such that every subgraph of Gp isomorphic to H is component of Gp. In

4.1 Strictly Balanced Graphs

83

particular,the number of components of Gp isomorphic to H tends to P;,, where 2 = ck- 1/a. Proof Since there are (k±+1)k-1 labelled trees of order k+ 1, the probability that Gp contains a tree of order k + 1 is at most

(k

) (k + l)k-lpk

=O(n-1/(k-l)).

Corollary 4.7 Let k > 3 be fixed, c > 0, p = cn-k/(k-I) and 2

ck-lkk-2/k!

Denote by X the number of trees of order k in Gp, by C the number of components of Gp with k vertices and by T the number of isolated spanned d

d

d

trees of order k. Then X-- P, C* Pý, C -+ P, and P(X = C = T)

--

1.

The proof of Theorem 4.1 and Theorem 1.21 imply easily that if H1 ,...,Hm are non-isomorphic strictly balanced graphs having the same (average) degree then in an appropriate range of p the multiplicities of the Hi's are asymptotically independent Poisson r.vs. The easy proof is left as an exercise. Theorem 4.8 Let H 1 ,..., Hm be strictly balanced graphs. Suppose Hi has ki vertices, li Ž_ 2 edges, and the automorphism group has ai elements. Denote by Xi = Xi(G) the number of subgraphs of G isomorphic to Hi. If the Hi's 2 have the same average degree, say 2k,/ll ..... km/lm = 2k/l, c is a positive constant and p -,'cn-k/l, then d

(X 1 ,. .. ,Xm) -d(PI ...... P2,), where 2- = cl'/ai, that is m

P(Xi =s,...,Xm = Sin)

-H

7-

/

for all si >_ 0,...,sm > 0. Since cycles are strictly balanced and have average degree 2, we have the following corollary. Corollary 4.9 Denote by Xi = Xj(G) the number of i-cycles of G. If c > 0 and p - c/n then P.....X.)......P, .....d where 2j

=

c1/2i.

84

Small Subgraphs

Proof The automorphism group of a cycle of length i has order 2i.

E

It is interesting to contrast this corollary with Corollary 2.19, giving the distribution of short cycles in regular graphs. In many ways a random r-regular graph is rather close to a random graph with rn/2 edges and to a random graph with probability r/n of an edge. However, in f-r-reg the number of i-cycles has asymptotically Poisson distribution with mean (r - 1)'/2i, while in the latter two models the mean tends to ri/2i. The condition of regularity makes the appearance of short cycles less likely. The formulae in the proof of Theorem 4.1 tell us that E(X)r is a fairly good approximation of Er(X). Therefore, in a suitable range of p, Theorem 1.22 enables us to deduce normal convergence. The result below is an extension of a theorem of Schfirger (1979b), who proved the result for complete subgraphs. Theorem 4.10 Let H and X = X(G) be as in Theorem 4.1. Suppose p p(n) is such that pnk/t

=

00o

and

pn/

0

for every e > 0. Then

(X -A) -

d

N(O, 1),

where An = (n)kp' /a.

Proof By (4.1) the expectation of X is exactly imply that A, -

An.

The conditions on p

oo and An = o(n') for every e > 0. We wish to apply

Theorem 1.22 to obtain normal convergence, so to prove our theorem it suffices to show that EJ(X)/2I

= 1 + o(Al-t)

(4.5)

for all r, t e N. Let r and t be fixed natural numbers and r > 0. By (4.2) we have E;(X) = Arn(n)rk { (n)k }-r = 2r(1

-

(4.6)

where 0 < Cn < (rk)2 /n = o(An t ).

(4.7)

4.2 Arbitrary Subgraphs

85

Furthermore, by (4.4), rk--1

Er (),n

r

/

)r

pl

pl/k En

rk)rks

rk- 1 s=k

<

2 p1/k

ýns-rksrkp(s/k-r)I s=k rk-1

>2,s/k-r as/k-rsrk-

,1/4 g-

plk)

s=k

=

Since Er(X) (4.8).

=

o(A).

(4.8)

E•(X) + E7(X), relation (4.5) follows from (4.6), (4.7) and D

In §3 we shall prove a considerably stronger result about Poisson approximation.

4.2 Arbitrary Subgraphs If H is not strictly balanced, the distribution of the number of Hsubgraphs is less straightforward. In order to gain some insight into the general case, we shall introduce a certain rather simple decomposition of a graph called a grading, whose definition is a little cumbersome. Suppose H is a proper spanned subgraph of F with F having k more vertices and I more edges than H. Then we define the additional degree of F over H as d(FIH) = 21/k. The maximal additionaldegree of F over H is m(FIH) = max{d(F'lH) : H : F' c F, V(H) :# V(F')}. To define a grading of a graph F we proceed as follows. Let F1 be a minimal subgraph of F with d(F 1 ) = m(F). Suppose we have defined subgraphs F1 cr F 2 (- ... ( Fk. If Fk = F we terminate the sequence. Otherwise let Fk+j be a subgraph ofF with V(Fk) c V(Fk+l), V(Fk+l) # V(Fk), such that d(Fk+IlFk) = m(FIFk). This increasing sequence of subgraphs has to end after finitely many terms, say in F, = F. Then we call (FI, F 2.... F5 ) a grading of F. For example, the graph F of Fig. 4.2 has grading (F1 , F2, F 3), where F1 is spanned by the vertices 1,2,3,4, F 2 is spanned by the vertices 1, 2,..., 9 and F 3 = F.

Small Subgraphs

86

12

11

3

7 10

2

5

6-

1

8

9

Fig. 4.2. A graph F with m(F)

=

3 and d(F) = 17/6.

Note that we have d(FI) = m(F) > d(F 2 1F1) > d(F 3 1F2) >... > d(FsiFs-).

In order to find an F-subgraph of a random graph G, we shall consider a grading (Ft F2, ... F,) of F. We shall find a copy of F, in G, then a copy of F 2 containing the copy of F 1, and so on. We shall need a technical lemma enabling us to find a copy of Fi+,, provided we have found a copy of Fi. To simplify the notation let FO be a spanned subgraph of a graph F such that F has k >- 1 more vertices and I more edges than F0 . Suppose F is the only subgraph of F with maximal additional degree over Fo :rm(FIFo) = d(FIFo) = 21/k. Lemma 4.11 Let V = V1 U V2 , V1 f V2 = 0, V2 1 = m(n) -* oo and pmk/I --, oo, and let Fo be a fixed graph isomorphic to Fo with V(Fo) C V1.

Then lim P(G zD H = Fo for some H -- FIG[V1 ] = Fo) = 1.

n--Ioo

Proof Let us fix a graph G1 = F0 with V(G 1 ) = V1. It suffices to show that lim P(G = H

D

FPo for some H -- FIG[V1 ] = G1) = 1,

so in the proof we shall take everything conditional on G[V1] = G 1. Denote by Y = Y(G) the number of F-graphs H in G such that H z F0 , V(H) - V(Fo) = V2 and there is an isomorphism between H and F carrying F0 into F0 . The (conditional) expectation of Y is clearly

E(Y) - c

(k

='it

4.2 Arbitrary Subgraphs

87

where c _>1 depends only on the pair (F, F0 ). Since pmik/1 -) o, we have p--* o as m --+ c. Let us give an upper bound for the second factorial moment E 2 (Y), that is for the expected number of ordered pairs (H 1, H 2 ) of appropriate H-graphs. Denote by E•(Y) the expected number of pairs (H 1 , H 2 ) with V(H 1 ) n V(H 2 ) = V(Fo) and set E"'(Y) = E 2 (Y) - E'(Y). Clearly

Furthermore, as in the proof of Theorem 4.1, if H 1 U H 2 has s < 2k - 1 more vertices than F0, then Ht U H 2 has at least sl/k + 1/k more edges than F0 . Consequently the following crude estimate holds: 2k-1

c Z

E it( )

S sl/k+l/k

s=k 2k-I

•

C2/t2 E

ms--2kpsl/k+l/k-21

s=k SC3 22m- Ip (Zk-1)1/k+l/k-21 =

pI c3 /22m-

2 = o(/2 ).

-1)k

Here c1 , c 2 and c 3 depend only on F and F0 , and the inequalities hold if m is large enough. Thus E 2 (Y) = { + o(t)}p

2

and E(Y 2 ) = {1 + 0(1)}/p2 . By Chebyshev's inequality we have P(Y = 0) = o(1), completing the proof,

E

This lemma enables us to prove the following extension of Theorem 4. 1. Theorem 4.12 Suppose the graph F has a unique subgraph F 1 with d(F1 ) = re(F) = 21/k, where k = IFI and 1 = e(FI) > 2. Let p - cn-k/I, where c is a positive constant and denote by Y = Y(G) the number of Fi-graphs d

contained in F-graphs of G E ý§(p). Then Y -- P (), where 2 = c'/a and a is the order of the automorphism of F1 . In particular,the probability that a r.g. Gp contains at least one F-graph tends to 1 - e-).

88

Small Subgraphs

Proof Let (F1,F 2,..., F,) be a grading of F, say with Fi having ki more vertices and li more edges than Fi- 1. By assumption Ilk > 12 /k 2 > 13 /k 3 ... Ž 1I/ks. Let 0 < e < 1/s and set c, = {1-(s)-1I)e/,= c1/a. Partition V into disjoint sets V1, Y2,..., V, such that

IVi Y

n

{I

Note that p ,- c^n7

1

-

(s

-

1)e}n and ni

=

Vil

en,

i

=2,3...,s.

, so by Corollary 4.5

P(G[V1 ] contains exactly r disjoint Fl-graphs)-

e-:t/r!. with

Since pn-ki/ti --* oo for i = 2, 3,...,s, for every fixed F1 -graph vertex set in VY we have P(G[V1 U 12] : H : F1 for some H 2 F 21G[V 1 ] DF1 ) -' 1. Similarly, if F, is a fixed Fi-graph whose vertex set is contained in then

I::

YLJj DH:DFý for some H 2tFi 1 G

P (G

L YJ

U>.=1 Vi,

- Pi)

Hence almost every graph containing r disjoint Fl-graphs contains r F-graphs whose Fi-subgraphs are disjoint. Therefore lir P(Y > r) > e-A

Z2{/j!.

n---co

r

Letting r tend to 0 this gives 0C

lim P(Y > r) > e-;Zi•J/j!, t--+ei

(4.9)

j=r

2 as e -+ 0. On the other hand, if X = X(G) is the number of F 1-subgraphs of G, then by Theorem 4.1 since 2,

-

0o

lim P(X > r) = e-ý E 2/j!. j=r

(4.10)

As Y(G) < X(G) for every graph G, relations (4.9) and (4.10) imply lim P(Y > r) = e-

n--

2J/j!,

oo

j=r

so Y

d

P(2), as claimed.

4.2 Arbitrary Subgraphs

89

In the result above we cannot discard the condition that F1 is the unique subgraph of maximal average degree. For example, if F = 2K 3 , that is F is a disjoint union of two triangles and X(G) is the number of F-graphs in G, then with p = c/n we have lim P(X=s)= 'e- 2r/r!' if s=

2

otherwise,

-0,

where 2 c 3/6. However, given an arbitrary graph F, one can determine the rough order of p that guarantees that a.e. G e W(p) contains F. -

Theorem 4.13 Let F be an arbitraryfixed graph with maximal average degree 21/k > 0. Then lim Pp(G =) F) n-oo

pni

if pnk/I

0,

L101, if pnk/1 pnk/

cc. 00o.

cc. Let F 1, F2 where ni =Vi-

... cz F, be a grading n/s, i 1.... s. Then

lim P(G[Vi] = F1 ) = 1.

(4.11)

Proof Suppose first that pnk/1

ofk11 F. Partition V as V

0',

=

--

U'=, Vi

oo, so by Theorem 4.1

Furthermore, by Lemma 4.11,

lim P (G

['.' ii -/ Fj+1 G [Lj Vii

F) =1.

(4.12)

Relations (4.11) and (4.12) imply lim P(G = F,) = 1. n---f2

On the other hand, if pnk/1

-*

0 then, again by Theorem 4.1,

lim P(G = F) :< lim P(G

n--oo

n---0

D F1 )

= 0.

Recalling the close connection between WP and gives us the threshold function.

SfpN],

Theorem 13

Corollary 4.14 The thresholdfunction of the property of containing a fixed non-empty graph F is n2- 2/m, where m = m(F). Li

90

Small Subgraphs

The approach taken above is somewhat of an overkill: the grading allows us to deduce much about the distribution of H-subgraphs of random graphs. However, if our aim is to prove Theorem 4.13 then, as shown by Rucifiski and Vince (1985), one can proceed in a much simpler way. To present this elegant proof of Rucifiski and Vince, write XH(G) for the number of H-subgraphs of a graph G, and for a non-empty graph F define tFF(n,p) = min{Ep(XH(Gp)) : H c F,e(H) > 0}. A moment's thought tells us that OF(n,p) • min{nIHI pe(H) : H ( F,e(H) > 0}, and so FF(n,p) --+ oO iff npm(F)/2 __ 0. More importantly, Rucifiski and Vince (1985) showed that 2(XF)F)

where the constants implicit in depend on F but not on n or p. Now, to deduce the essential part of Theorem 4.13, note that if npm(F)/ 2 __ c then Pp(Gp ý F)

Pp(XF

a2(XF)

- 2 0) 5 (E(XF))

1F

so a.e. Gp has an F-subgraph, as claimed. Karoiski and Rucihski (1983) proposed yet another way of proving Theorem 4.13. They conjectured that every graph F can be extended to a balanced graph H with the some maximal average degree: there is a balanced graph H containing F with m(H) = m(F). Combined with the result of Erd6s and R~nyi (1960) that Theorem 13 holds for balanced graphs, the conjecture implies Theorem 13. This conjecture of Karofiski and Rucihski was proved by Gy6ri, Rothschild and Rucifiski (1985) and Payan (1986). In fact, it need not be very easy to find an appropriate balanced extension: Rucifiski and Vince (1993) showed that the balanced extension may have to have many more vertices than the original graph. The appearance of F-subgraphs in random graphs was studied in great detail by Janson, Luczak and Rucifiski (1990). Among other results, they proved the following beautiful inequality involving the probability Pp(Gp : F) and the function OF(n,p). Theorem 4.15 Let F be a fixed non-empty graph and, as before, set

4.3 Poisson Approximation

91

4)F(n,p) = min{Ep(XH(Gp)) : H = F,e(H) > 0}. Then e-4/(1-P) _
• F) < e-c•Dv

for some positive constant c = CF.

0

The real content of this theorem is the upper bound; it can be proved by making use of the martingale inequality given in Theorem 1.20. For a sketch of this proof and many other beautiful results about small subgraphs of random graphs, see Chapter III of Janson, Luczak and Rucifiski (2000). 4.3 Poisson Approximation Let us return to the study of strictly balanced subgraphs of random graphs. Our aim is to approximate the distribution of the number of subgraphs isomorphic to a fixed strictly balanced graph. In §1 we gave such approximations and found our task fairly easy provided the expectation is bounded or grows rather slowly with n. In this section we shall present a theorem of Barbour (1982) which is considerably stronger than the results of §1, and permits much faster growth of the expectation. Theorem 4.16 Suppose H is a fixed strictly balanced graph with k vertices, 1 > 2 edges and with a elements in its automorphism group. Denote by Y = Y(Gp) the number of H-subgraphs of Gp. Suppose that , = (n)k p 1 00 a and p =o(n

k/{l+1/(k-2)}).

Then the total variation distance between the distribution of Y and Pý, tends to 0, in fact

d{jY(Y),Pj} = O(nk-2p k-2)1/k+/k)

=

O(1).

(4.13)

Proof Let -Y be the collection of all H-graphs whose vertex set is contained in V. Then

(n)k a

IJfI= (n) kJ[ k! a and E(Y) =

pI•'

=

4n.

92

Small Subgraphs

With a slight abuse of notation the elements of A will be denoted by a, l,3... For o E and Gp G ýVp set ,= XG

=

, X0,

so that Y

=

>

ifa c=

,

otherwise

Xc, where the summation is over *. Furthermore, put Y

n=. Xfl,

where fln denotes the set of common edges of the graphs /3 and L, and the summation is over all elements #3of - satisfying #fln a 0. Finally, let us write p•, for E(X,,) = pt. Our aim is to apply Theorem 1.28 of Chapter 1 to estimate the total variation distance between the distribution of Y and the Poisson distribution. The following identity, valid for all functions x : Z+ --+ R, will enable us to estimate the left-hand side of the equation appearing in Theorem 1.28(iii): A2x(Y + 1)- Yx(Y)

Zp

+ 1)-x(Y.+ 0)} +{x(Y

oc

"+ '(p - X)x(Yc + 1) "+E X,{x(Yc. + 1)- x(Y)}.

(4.14)

cc

Now let A be an arbitrary subset of Z+ and let x = XA = X;,,A be the function defined in Theorem 1.28. Since Y = Y, + ±flnc.7z XP 3, lx(Y + 1) - x(Yc. + 1)•< (Ax)

Y, X# . 3

(4.15)

X#.

(4.16)

Furthermore, if X, = 1 then 0_ Y - Y" - 1 =

xf, x#

so

X, {x(Y., ± 1)- x(Y)}ý < (Ax)X. E fl3 c.

4.3 Poisson Approximation

93

Let us take the expectation of (4.14). Then (4.15), (4.16) and the independence of X, and Y, imply that

E{2nx(Y±+1) -Yx(Y)}j

E p,(Ax)E

X#

I.f

+±(Ax)ZE E

p~pp +-~

•(Ax)

E(XoXfl)} (4.17)

# 0,

If # nl implies

then IV(/#)

V(c•)j > 2 so the deduction of relation (4.4)

z E(X,,Xf) 0 (E~ =

flBn•

Since nkpl

-I

nSp sllk±/k)

\s=k+l

o, the sum above is O(n2k-2p(2k-2)l/k+I/k).

(4.18)

Also,

Z

PP

= O(1Yflplnk- 2 pI) = O(n2k-2p21) =

O(n 2 k- 2 p(2k-2)/k+l/k).

(4.19)

Therefore Theorem 1.28(ii) and relations (4.17), (4.18) and (4.19) imply

E{j.nx(Y + 1) - Yx(Y)}I

A.O(n2k-2p(2k-2)lk+l/k)

•

2

-

O(nk-2p(k- 2)l/k+l/k).

(4.20)

Since the constants hidden in the O's above are independent of the set A, Theorem 1.28(iii) and (4.20) give d{ýY(Y),P,} = sup E{,,x(Y ± 1)- Yx(Y)}J - O(nk-2p(k-2)l/k+l/k), A

as claimed by (4.13).

Small Subgraphs

94

Exercises 4.1

What is the threshold function of the property of containing a cycle? And a cycle and a diagonal?

4.2

Use Theorem 1.21 to prove Theorem 4.8.

4.3

Let F be a fixed graph with maximal average degree m > 0. Show that if lim, P(GM : F) = 1, then M/n 2 - 2/n - oc and if limn P(GM = F) = 0, then M/n2- 2 /m --_ 0.

4.4

Given e > 0 and 0 < x < 1, find a first-order sentence Q and a sequence {M(n)} such that n2-e < M(n) • N/2 and lim PM(Q) = x. (cf. Theorem 2.6. Let Q be the property of containing a large complete subgraph.)

4.5

Let U and W be disjoint sets with Ul = m, W1 = n. Take a random bipartite graph B 11 2 with vertex classes U, W and with probability ½ of an edge. Given B 112 , let Tm = T(B11 2 ) be the directed bipartite graph with vertex classes U and W, in which uw is an edge if uw e E(B 112 ), and i~u is an edge if uw ý E(B1/2), u E U, w E W. We call Tn,m a random bipartite tournament. Denote by Xn,m = X(Tn,m) the number of oriented 4-cycles in Tnm. Use the method of Theorem 2.15 to show that 8Xn,_ _ (mn) (n) (;)

(n)

(2m + 2n

-

1/2 , N(0, 1).

[Moon and Moser (1962); see also Frank (1979b). The probability Pi,n that a tournament contains no 4-cycle (i.e. it is acyclic) was estimated by Bollobfss, Frank and Karofiski (1983), who also calculated the asymptotic value of Pn,[cnj. In particular, they proved that

P,,n - 2 -

2

(n !)2 (log 2)-2n+1 {nr(1 - log 2)}1/2.

The estimate above is connected to the enumeration of restricted permutations and to estimates of Stirling numbers; see, for example, Lovfsz and Vesztergombi (1976), Szekeres and Binet (1957) and Vesztergombi (1974).]

Exercises 4.6

95

Let T be a tree with vertex set V(T) = {X, .... Xk}. Furthermore, let 0 < c < o and p - c/n. Write Y = Y(Gp) for the number of components C of Gp such that V(C) = {yl .... ,Yk} for some 1 _ Yl < Y2 < ... < Yk <--n and xi*+-y gives an isomorphism between T and C. Prove that E(Y) Tyon for some To > 0 and r2(Y) = O(n). Deduce that Y(Gp) = yon + o(n) for a.e. Gp.

5 The Evolution of Random Graphs-Sparse Components

In the next two chapters we shall study the global structure of a random graph of order n and size M(n). We are interested mostly in the orders of the components, especially in the maximum of the orders of the components. The chapters contain the classical results due to Erd6s and R~nyi (1960, 1961a) and the later refinements due to Bollobis (1984b). Before getting down to the details, it seems appropriate to give a brief, intuitive and somewhat imprecise description of the fundamental results. Denote by Lj(G) the order of the jth largest component of a graph G. If G has fewer than j components, then we put Lj(G) = 0. Furthermore, often we write L(G) for LI(G). Let us consider our random graph GM as a living organism, which develops by acquiring more and more edges. More precisely, consider a random graph process G = (G,)0. How does L(GM) depend on M? If M/n(k-2)/(k-1) -, oo and M/n(k-])/k -* 0 then a.e. GM is such that L(GM) = k. If M = [cn], where 0 < c < ½,then in a.e. GM the order of a largest component is log n but if M = [cn] and c > 12 then a.e. GM has a unique largest component with about en vertices, where E > 0 depends only on c. (The unique largest component is usually called the giant component of GM.) Furthermore, in the neighbourhood of t = n/2 for a.e. G = (Gt)' the function L(Gt) grows rather regularly with the time: L(Gt) increases about four times as fast as t. The present chapter is devoted to the study of components which are trees or have only a few more edges than trees. The main aim of the subsequent chapter is to examine the giant component. 5.1 Trees of Given Sizes As Components In Chapter 4 we determined the threshold function of a given tree. If T is a tree of order k whose automorphism group has order a, 96

5.1 Trees of Given Sizes As Components

97

M = M(n) - !cn 2-k/(k-I) for some constant c > 0 and X 1 denotes the number of subgraphs of a random graph GM isomorphic to T, then X- d P(A),

where A = ck-I/a. Also, if X 2 denotes the number of trees of

order k contained in a graph GM, then X 2 d- P(y), where p = ck-1kk- 2 /k! Furthermore, if X 3 is the number of components of GM that are trees of d

order k, then X3 -* P(p) also holds. As M(n) grows to become of larger order than n2-k/(1-1), the number of tree components of order k of a typical GM tends to infinity with n. However, in a large range of M(n) the distribution of the number of tree components of order k is still asymptotically Poisson. The proof of this beautiful theorem of Barbour (1982) is rather similar to the proof of Theorem 4.15, in particular, it is also based on Theorem 1.28. To simplify the calculation we work with the model #(n,p). Let p = p(n), k = k(n) and denote by Tk = Tk(Gp) the number of isolated trees of order k in Gp. Write A for the expectation of Tk. As by Cayley's formula there are kk- 2 trees with k labelled vertices, =E(Tk) = (n)

(5.1)

kk-2pk-1 qk(n-k)+(')-k+l

Theorem 5.1

dY_(Tk),P)}

• 2{cl(n,k,p) + c 2(n,k,p)},

where

1) (lnk~)=

c1(~k(n=

-

k-

2

_k-I

qk(n-k)

T( ) )k -k i

n

and c2 (n,k,p) = k~p (n

k) kk2pk

qk(n2k)+(•)k+1

= (n -k)kk2pq-k2A < e-k 2/nk2pq-k2 . (n)k Proof Let Jf be the set of all trees of order k whose vertex set is contained in V. As in the proof of Theorem 4.15, we shall use c•,fl, 7,... to denote elements of Y. Furthermore, we shall frequently suppress the suffix k in Tk. For o( E Y and Gp E SP set X,(G)

=

1, if a is an isolated tree of Gp, 4G=0, otherwise.

The Evolution of Random Graphs-Sparse Components

98 Then °-I

qk(n-k)+(k)-k+ I =

P(X=l)

,EX)

(n)k

and

E(T) ý_E(Tk)

=

=SE(X,)

(n) kk- p 2

Let T* = T*(Gp) be the number of isolated trees of order k in the graph Hp = Gp[{1,2,..., n - k}], the subgraph of Gp spanned by the vertices 1, 2,..., n - k. Then for all x : Z+ -. IR and a e we have .F

E{Xx(T)} = P(X, = 1)E{x(T* + 1)}.

Hence E{)x(T + 1)-

Tx(T)}

=

XE{x(T + 1,}-- EE{X~x(T)}

= ;E{x(T + 1)} -

--

P(X• = 1)E{x(T* + 1)},

AE{x(T + 1)--x(T* + 1)}.

Furthermore, since ]E{x(T + 1) -- x(T* + 1)}1 < (Ax)E(]T - T*), we have ]E{2x(T + 1) - Tx(T)}j I! 2(Ax)E(jT - Tr*).

(5.2)

In order to be able to apply Theorem 1.28, we need an upper bound of the right-hand side of (5.2). Clearly T-T*

<_

n

Z j,

j=n-k+l

where 1, if the vertex j is in an isolated tree of order k in Gp, {0, otherwise. Therefore E{max(T-T*,O,}) _

E(Zj) < k

(

-)

kk- 2p= cI(n,k,p). (5.3)

j=n-k+l

On the other hand, T* - T is certainly not larger than the number of

5.1 Trees of Given Sizes As Components

99

isolated trees of order k in Hp which are joined to some vertices among n - k + 1...,.n. Consequently E{max(T* - T,O)} _ (I - qk 2)E(T*) • k 2p (n

-

k)

kk-2pk-qk(n-2k)+()-k+l =

c 2 (n, k,p).

(5.4) Now let A = Z+ and let x = XAA be the function defined in Theorem 1.28. Then by Theorem 1.28 and relations (5.2), (5.3), (5.4) we have IP(T e A) -P(A)ý

<- 22min{1,ý-l}{c 1 (n,k,p)+C 2(n,k,p)}

• 2{c 1(n,k,p) + C2(n,k,p)} and so d(Y(Tk),PA) :5 2{c(n,k,p) + c2(n,k,p)}. Corollary 5.2 If p(n)k(n) = 0(1) as n --+ oo, then

d{Y(Tk),Pý} - 0 provided any of the following three conditions is satisfied: (i) k(n) --* oo,

(ii) np(n) --* 0,

(iii) k(n) Ž 2 and p(n) = o(n- 1).

Proof Check that the conditions imply that c 1(n,k,p) +c 2 (n,k,p)

=

o(1)

(Ex. 2).

El

In particular, if k > 2 is fixed and p = o(n- 1), then Tk has asymptotically Poisson distribution. As M(n) continues to grow and M(n) = o(n) no longer holds, then Tk ceases to be asymptotically Poisson. However, as Corollary 5.2 shows, Tk again becomes asymptotically Poisson when M(n)/n -+ oo. Furthermore,

E(Tk) becomes bounded away from both 0 and o if M(n) = (n/2k){logn + (k - 1)loglog n + ca(n)},

where a(n) is a bounded function. This can be read out of Corollary 5.2, but it is also a simple consequence of Theorem 1.20. Theorem 5.3 Let p = {logn+(k- 1)loglogn+x+o(1)}7kn, where x E R is fixed and denote by Tk the number of components in a random graph Gp d

that are trees of order k =: 1. Then Tk --+P(2), where

,=e-x/

(k -k!).

100

The Evolution of Random Graphs-Sparse Components

Proof As by Cayley's formula there are kkEr(Tk) =

trees with k labelled vertices,

.. (n -(r~ l)k)

(n) (n -k)

x(

2

k-)p~

P)rk~n-rk)

-1

+

(')

r(k

-

1).

Indeed, in order to guarantee that r trees of order k with disjoint vertex sets are components of a graph Gp we have to make sure that r(k - 1)

given pairs are edges and rk(n - rk) + ( k) _ r(k-

1) other pairs are

non-edges. The assumption on p implies Er(Tk)

,

' knr kk2)rprk-1)(l _- p)rkn (k!)r

(knk~-2 (kk-2

I n-(k-1)Ilog log n--1eog

kn

n(log n)k

The result follows by Theorem 1.21.

)

l

We can now give a complete description of the asymptotic behaviour of the number of isolated trees of order k. Theorem 5.4 Denote by Tk the number of components in a random graph GP that are trees of order k >Ž 2. (i) If p = o(n-k/(k- 1 )), then Tk = O for a.e. Gp.

(ii) If p - cn kIk-1). for some constant c > 0, then Tk -* P(cklkk-2/k!). (iii) If pnk/(k 1) _, oc and pkn- logn- (k-- 1)loglogn -+ -co, then for every L E lR we have lim P(Tk > L)

= 1.

n-oo

(iv) (v)

If pkn-logn-(k-1)loglogn -* x for some x E R, then Tk d pieX/ (k. k!)}. If pkn - logn - (k - 1)loglogn -* oo, then Tk = Ojor a.e. Gp.

Proof Parts (i) and (ii) were proved in Chapter 4, and (iv) is Theorem 5.3, above.

5.1 Trees of Given Sizes As Components

101

To see (iii) suppose pkn < 2k log n. Then, as in the proof of Theorem 5.3, n-nk-k--2k1-

E(Tk)

p)kn

k! kk-2 nkpk-1 e-Pkn

kk-2 n(pnekpnkJ))k-.

The last expression increases as pn increases from 0 to (k - 1)/k, and from then on it decreases as pn increases. Hence if pnkl(k- 1) -- o and pkn - logn - (k - 1)loglogn --- -oo, by parts (ii) and (iv) we find that E(Tk) --+ oc. Furthermore, 2k

E2T

-k)

I n 2k p2(k-1)(1 - p)kn - E(Tk)2,

so the assertion follows by the Chebyshev inequality. If p is rather large, say p Ž_ 3(log n)/n, then a.e. Gp is connected (Ex. 14 of Chapter 2), so Tk 0 for a.e. Gp. Finally, if pn _•<3 log n and pkn - log n - (k - 1) log log n - oo then, as before, kk-2

E(Tk)

0,

n(pnekPn/(k-))k-I

implying (v). The proof above shows that for a fixed k the expected number of isolated trees of order k in GP increases as p increases up to p - (k-1)/kn, and from then on it decreases. Furthermore, the maximum is about (k -

I)k-1

k .k !ek-I

For large values of k this is about (1//2m)k- 5/2 n. To conclude this section, we state two beautiful theorems of Barbour (1982) about approximating Tk by the normal distribution. The first result concerns the space W(n, c/n), the space we shall study in this chapter and the next. Theorem 5.5 Let k > 2 and c > 0 be fixed and let p = p(n) - c/n. Then sup

]P[(Tk

--

k)/0k •

x] -

(I(x)l =

0(n 1/2),

x

where )Lk

=

E(Tk)

- nkk-

2

ck-le-kc/k!

102

The Evolution of Random Graphs-Sparse Components

and 2

2 = O (Tk) O-k 0'+

+ (C

_n{1

-

(

1)(kc)k-ekc/k!}. ' 1 1(

:

It is clear that for c @ 1 Poisson convergence does not hold, namely d{Y(Tk),Pý, } 0. Theorem 5.6 For every fixed k >_2 there exists a constant C(k) such that the r.v. Tk on ýI{n,p(n)} satisfies sup P{(Tk x

- x} -- 0(x)0 • C0x j1

- 2k)/6k

if n is sufficiently large, where

E(Tk) and Uk

Xk

= a(Tk).

5.2 The Number of Vertices on Tree Components In the previous section we studied the distribution of trees of a given order. Our main aim in this section is to obtain information about the total number of vertices on components that are trees. This information will be crucial in the study of the order of a largest component. Denote by T the number of vertices of Gp belonging to isolated trees: T = ZkikTk. The next result shows that if p = o(1/n) or p ,- c/n for some c < 1, then almost all vertices belong to isolated trees, but there is an abrupt change at p ,- 1/n. Theorem 5.7 (i) If p = o(1/n), then T = n for a.e. Gp. (ii) Suppose p = c/n, where c is a constant. IfO< c < 1, then

E{T(Gp)} = n + 0(1) and if 1 < c < oo, then E{ T(Gp)} = t(c)n + 0(1), where t(c) = 1

k! (cekk

]

k~l

Remark. By standard but tedious manipulations one can show that the function s = s(c) = ct(c) is the only solution of se'

= ce"

(5.5)

5.2 The Number of Vertices on Tree Components

103

1.0 0.8 0.6 0.4 0.2 0.0

0

1

2

3

4

Fig. 5.1. The curve y = t(c). in the range 0 < s _<1. In particular, t(c) = s(c)/c = 1 for 0 < c •< 1

(5.6)

and part (ii) of Theorem 5.7 states simply that E(T) = t(c)n + 0(1)

(5.7)

whenever p = c/n and c € 1. In the proof below we shall show (5.7) instead of the assertion in part (ii). Since E{T(Gp)} is a continuous function of p, Theorem 5.7 implies that E{T(G 1 lj)}

=

n + o(n).

(5.8)

It should be stressed that (5.8) cannot be strengthened to E{T(Glln)} = n + 0(l). Relation (5.6) can be shown without any analytic manipulations. It is easily seen (cf. Ex. 8) that if p - c/n and 0 < c < 1, then the expected number of vertices of GP on components containing cycles is 0(1), so the expected number of vertices on tree components is {1 + o(1)}n. Hence if (5.7) holds as well, as we are about to show, then Cfkkl1

k!

(ce-c)k=c if

0<< c1.

k=1

This shows that s = s(c) = ct(c) is a solution of se-s = ce-c in 0 < s < 1. It is easily seen that there is only one solution, so s is exactly this solution.

104

The Evolution of Random Graphs-SparseComponents

Proof of Theorem 5.7 (i) Write Xk for the number of k-cycles in Gp, where p • c/n. Then 2

2k

so cfC

ciD

E E(Xk) <

E ck.

k=3

k=3

Hence if p = o(1/n), almost no Gp contains a cycle, that is a.e. Gp is a forest. (ii) Suppose p = c/n where 0 < c < o and c :ý 1. We shall give a particularly detailed proof of the assertion, because the proof of a slightly weaker assertion in Erd6s and R6nyi (1960) is based on some incorrect estimates. Clearly E(Tk)

= (n ) kk-

2

(C)-

4

(1--

C)kfn-k(k+3)/2+1

(5.9)

Hence if 1 < k < n1/2, then bk-2

2

E(Tk) < n k!- ck-1 eck eck /(2n)+3ck/(2n) k-2

< n •k

c-eck(1

ck 2 /n)

(5.10)

and

kk- 2 ( 1- k)k ck-

>

E(Tk)

2 ecke-c k/n

> n k! ck-le-ck(1--c2k 2/n) where cl and c2 depend only on c. Consequently if 1 < I < m then E (mkTk)

(5.11) n

kk-k-ck

-

Skkk+l <

max{eC,c

2

}m

k lck-lek = 0(1).

(5.12)

k=l

The assertion of our theorem will follow from inequality (5.12) if we

5.2 The Number of Vertices on Tree Components

105

show that the terms with k > n1/ 2 contribute rather little to the sums. Note that by relation (5.9) that if 1 < k < n, then 1+

E{(k+l)Tk+l}/E(kTk)=(n--k)

C

)-k2

-

k n

(i i)ece-'(1

k/(lE)

n

)flAk2

<

>

(5.13)

since xel-x < 1 for x > 0. Set k, = [nl/ 3J. Then by Stirling's formula E(kl Tk,) = o(n-M) for every M > 0, and so (5.13) implies kTk ) •E(klTk,)Z2j = O(n 2 )E(klTkI) = o(n- 3 ).

E (

(5.14)

j=0

(k="ki

The tail of the series defining ct(c) is also rather small: _

kk

kkI (

: k 3 / 2 (celc)k

1

, - kk=k

o(n1)

(5.15)

1

since cel-c < 1. Relations (5.12), (5.14) and (5.15) imply the desired estimate of E(T):

/k=1

\k=l

1n k! ck-le-ck k=l

+E+n (~kTk) k Tk \k=kl

1

k=

kTk) -

(

eck

kk-I

n

E (nkTk)

/

+ nny 00k k-

-

c

ek

o. ) k= O (1).

k=kl

Corollary 5.8 Suppose p = c/n, 0 < c < 1, and co(n) - oo. Then a.e. Gp is such that every component is a tree or a unicyclic graph, and there are at most (o(n) vertices on the unicyclic components. Proof We may assume that wo(n) • log log n. Since T(Gp) < n for every Gp and E(T) = n + 0(1), lim P{T <_ n-wo(n)} = 0.

n-oo

106

The Evolution of Random Graphs Sparse Components

To prove the second part it suffices to show that almost no Gp contains a component of order k < o)(n) with at least k + 1 edges. This can be done by the following crude estimation of the expected number of such components:

~k-112

,(n)(n)

-k~l

w(n)V

2 2/n

)<•:k2

n k=4

=

F~1

o(1).

k=4

In §4 we shall examine the components containing cycles in considerably greater detail. If p = c/n and c > 1, then T(Gp) is still concentrated close to its expectation, though it does not have such a strong peak. Theorem 5.9 Suppose p = c/n, c > 1 and co(n) -->. that

Then a.e. Gp is such

1T(Gp) - t(c)nl < wo(n)n 1 / 2. Proof Let kl = [nl/ 3 J. Then by relation (5.14) k.k> Ž5

P

0.

k=kl

Set TF

I kTk. Then by Theorem 5.7 JE(T) - t(c)nl = 0(1),

so it suffices to prove that P(IT - E(T)j > wo(n)n1/2 + O(1))

-

0.

(5.16)

For 1 < k < I < ki the expected number of ordered pairs of components of Gp, the first of which is a tree of order k and the second is a tree of order 1, is clearly

(n)(n-k k~ 2 l~ 2

< E(Tk)E(TI)

I -

(c)k+

1 2

-

(i-c)(k+l)(flk-1)T(k+7)-k1+2

C)-kn < E(Tk)E(Tl) (I + 2ckl.

Therefore if 1 < k < 1 < ki, then E(Tk T,) • E(Tk)E(T,)

I + 2ckl)

5.2 The Number of Vertices on Tree Components and if 1

107

k < k1 , then E(T•)

=

E{Tk(T -])If+ E(Tk)

< E(Tk) 2 (i + 2ck 2 ) + E(Tk).

Hence

E(T2

E

{ (

(k=1

:

Tk) 2 1 + 2Ck 2

_< k 2 E(Tk)2

+ E(Tk)

k=1

2ckl)

± 2ZEklE(Tk)E(Tl) (I+

S{E(k)}

2

E(Tk)

{k

n-

+ 0

k{E(Tk)

E(T)2 + O(n),

(5.17)

since ce1-C < 1 implies that implies

Z.-

1,k

2

E(Tk) = O(n).

Inequality (5.17)

6 2(t) = O(n) so by the Chebyshev inequality P {IT -E(T)> ! I )(n)n1/2

4 2(t)=

),

implying relation (5.16).

D

In the proofs of Theorems 5.7 and 5.9 we made use of the fact that 2 TZk=k, k Ep(Tk) = o(1) if p - c/n and c > 1. In fact, trees of considerably smaller order are also unlikely to appear in Gp. Theorem 5.10 Let 0
oo,c • 1 and p

=

c/n. Set a

=

c-l-

log c

and let ko=

Denote by X

-1

=

logn

-

loglogn

-

lo

EN,

lo

=

0(1).

X(Gp) the number of components of Gp that are trees

108

The Evolution of Random Graphs-SparseComponents

of order at least ko. Then X has asymptotically Poisson distribution with mean co - e-°do. 1 1t-e-• O5/2 cx/-7 Proof It is easily seen that for M > 0 there is a constant c2 > 0 such that

E

(1,

Tk)

where k2 = [c., log n].

o(nM),

(k=nk2)=

Hence it suffices to show that k2 Y =

Tk k=ko

has asymptotically Poisson distribution with mean Ao. Now

E(Y)

=

(n)

k

kk

2

(

(C)kkn

C kflk

2

/2+3k/2-l

k =ko k2

,,-

n

k-5/2(c el-c)k

E

cN/217 k=ko n u r c \/'27• 1=0

logn) -

(ko+5)

0 5i/2 e 10 cc

Furthermore, straightforward calculations show that for r > 1 the rth factorial moment of Y is Er(Y) = E(Y)r{1 + O(k r/n)} - E(Y)r _ A, Hence the assertion follows from Theorem 1.21.

El

Corollary 5.11 Let 0 < c < oo, c f 1,p = c/n, = c-- I - log c and wo(n) -* oo. Denote by L(t)(Gp) the maximum order of a tree component of Gp. Then a.e. Gp is such that L(t)(GP) -

a

I og n-

2

log logn

(n).

5.2 The Number of Vertices on Tree Components

109

The last result indicates that if p is very close to I/n, then the maximum order of a tree component of Gp is of larger order than log n. We shall see in the next section that as p approaches l/n, this maximum order becomes n2 / 3 . The study of the number of tree components is similar to the study of the number of vertices on tree components and the result is no surprise. The proof, which is very close to the proofs of Theorems 5.7 and 5.9, is left to the reader. Theorem 5.12 Let 0 < c < oo,c = 1,p = c/n and denote by U the number of tree components of Gp. Then

=

U(Gp)

E(U) = u(c)n + 0(1) where

u(c) = 1

c k

kk-2 k! (ec)k

and o' 2 (U) = 0(n). If ow(n)

-

c, then a.e. Gp satisfies

1U(Gp) - u(c)nl •! w(n)n1 2 .

U

1.0 0.8 0.6 0.4 0.2 0.0

4 42

0 Fig. 5.2. The function u(c).

Remark. Once again, by tedious manipulations one can show that u(c)= 1-c/2

for

0•
110

The Evolution of Random Graphs Sparse Components

This is also a consequence of Corollary 5.8 and Theorem 5.12. Indeed, for 0 < c < 1 a.e. G,1, is such that at least n - log n of its vertices belong to tree components. Since a.e. Gcl, has {c/2+o(1)}n edges, a.e. Ge/n must have {1 - c/2 + o(1)}n tree components. Hence u(c) = 1 - c/2 for 0 < c <1.

5.3 The Largest Tree Components For what function k = k(n) is it true that for some p = p(n) almost every Gp contains a component which is a tree of order k? For what ko = ko(n) is there some po = po(n) such that almost every Gpo contains a component which is a tree and has at least k0 vertices? The main aim of this section is to answer these questions. It will be convenient to write 0 = O(n) for np = np(n), so that p = O/n.

As we shall consider random graphs Go/n for various functions O(n), occasionally it will be necessary to write E01, instead of E, indicating that the expectation is taken in %(n,O/n). (Since O/n < 1, E01 n cannot be confused with Er, the rth factorial moment.) Let us recall (5.1), the formula for Eo/l(Tk), the expected number of trees of order k in Go/n: Eo/ (Tk) =

(n)

2

k-

()k"

(1

O)kn-k(k+3)/2+1

-

(5.18)

Theorem 5.13 Let 2 < k(n) < n. Then Eo/,(Tk) attains its supremum in 0 > 0 at Ok = (k - 1)n/[k{n - (k + 1)/2}]. If k = n then Ok = 2 and Ez/n(Tn)

(5.19)

- (e3/2n)(2/e)n.

Ifk < n- 1, then Eok/n(Tk)

=

0(1)

k(n

k)) 1/nk

k S

2

±1

) -k

k) n-k

kfn-(k/2)}

n - k/2

-nk

= O(nk 51 / 2 ). If k = o(n) and k

E

0MT)

Eo/nTk

-

(5.20)

oo, then

( 1

n

nk-2 12(n221k(n -k)) nk

1+ k 1 nk)

n-k

5.3 The Largest Tree Components

~< ~k 1+ <_n

(-

-

111

)

-k/

kn-k/2)

o(1)-5/2 k -5 2

( 5 .2 1)

Finally, if k = o(n 1/2 ) and k --+ oo, then nk-5/ 2

Eok/n(Tk)

(5.22)

El/n(Tk).

Proof The fact that Eo/n(Tk) attains its supremum at Ok = (k- 1)n/[k{n(k + 1)/2}] follows from (5.18) upon differentiating with respect to 0. Hence =

EOk/n(Tk)

(n) k k-

x Clearly On

=

2

-)k- (

( k

1-(k

n

)/2}

1-k{nki

k)

.

(5.23)

2 and 2/n(Tn)= nn-

2

(2)n-1

(1-2n)n(n-3)/2+i

e2 (2/e)' 2n

since (1-

2

•)

n /2

I

The first part of relation (5.20) is a consequence of (5.23) and the inequalities al/ 2 (a/e)a < a! < 4a'/ 2 (a/e)a,

which hold for all a >_ 1. The second part of relation (5.20) requires a little more work. We shall make use of the fact that if x > 0, then (1 + 1/x)X/ {e (I

2(x+ 1))} < 1.

(5.24)

Furthermore, since the left-hand side of (5.24) is a continuous function of x and tends to 2/e as x

(1 + 1/x)X/

->

{e

0,

2(x+

)

<

- 0 <1

(5.25)

112

The Evolution of Random Graphs Sparse Components

for all x,0 < x < 1, and some co > 0. Hence with x = (n - k)/k we find

that (±

<

{ + ( kn~

(i +

-k( kk

(ni

lkk/2

-

_k

k n-k)l

(1+1/~x1 - 2(x +1)1)) (I + 1/X

)

e-1

1

(5.26)

and if (n - k)/k < 1, that is k > n/2, then (

)n-k (n

k

(

k)k

EoA/n(Tk)

= O{n 3 / 2 k

)

• (1 -E)k <(1 -Fo)n/2.

Therefore 5 2

/ (1 _

0)n/21

if k > n/2, and otherwise Eok/ln(Tk) = O(nk-5/2),

proving the second part of (5.20). Relation (5.21) is an immediate consequence of (5.23), (5.26) and the Stirling approximation. Finally, if k = o(n1/ 2 ) and k

oo, then

--

(1+ n -

-)k

,ek,

iy Y

and n -- Ik12 )

implying our last assertion.

)

e-k'

El

Armed with Theorem 5.13 we can determine how small k(n) has to be if we want to guarantee that for some p almost every Gp contains an isolated tree of order k.

5.3 The Largest Tree Components Theorem 5.14 (i) If k/n2/ 5 tree of order k.

__

co, then almost no Gp contains an isolated

(ii) If c > 0 is a constant and k PI{c

5 2

113

'-'

2/5 cn2 5 , then in G11l

we have Tk

d

2

/ (2r)- 1/ 1.

(iii) If k/n 2/ 5 -, 0, then a.e. G1/, contains an isolated tree of order k. (iv) If ko/n2/7 --+ 0, then a.e. G1/l contains an isolated tree of order k for every k < ko. Proof (i) Suppose k/n

2

/5

--_

o and 0 < p < 1. Then by (5.19) and (5.20)

Pp(Tk > 0) •! Ep(Tk) •; EOk/n(Tk) = O(nk-5 1 2 ) = o(1).

(ii) Assume that k - cn2/ 5 . Then in <#(n, 1/n) the rth factorial moment of Tk is easily estimated: Er(Tk)-

(nrtk~k)f~ (kl (n)kkrk-2)n-rk-)

1nkO~

-

nrkO(k)

(k !)rn

{(n) k 2 n-k1 k

I

infk}ýr-IET~

Since E(Tk) - (1//2-r)c-51 2, the result follows by Theorem 5.20 of Chapter 1. (iii) Suppose 1 < k, _
(k)

(nl k X

(1 -

k

2

2

-

kk2

2

pkl+k2

p)ki{n-(ki+3)/2}+l+k 2{n-(k 2 +3)/2}-+-kik

2

so (1k

(n

Ep(Tk,,Tk2 )/Ep(Tk,)Ep(Tk 2)

p)-kk2

(n)k, <

(

)k,(

l

-klk2

since for 0 < a : 1 < b we have 1 -ab

(1 -

a)b.

Consequently Ep(TkI, Tk,) < Ep(Tk,)Ep(Tk2 ).

(5.27)

114

The Evolution of Random Graphs-SparseComponents

Now if k = o(n 2/ 5 ) then in 'ý(n, 1/n) we have E(Tk) (5.27), E(T2') = E(Tk, Tk) + E(Tk) < E(Tk)

2

--*

oo. Therefore, by

2 + E(Tk) - E(Tk)

and by the Chebyshev inequality P(Tk = 0) < E(T2')/E(Tk) 2

1

-

o(1).

=

(iv) Suppose k < k0 = o(n 2/ 7 ). Then in ý1(n, 1/n) E(rk) -' (n/Nff2_7)k-52 Furthermore, as in (iii), by (5.27) we have P(Tk =0) < 1/E(Tk). Therefore P(Tk = 0 for some k < ko) =0

n-1 Zks/} k=1

=

o(1).

)

Let us see now the maximal order of a tree component. Theorem 5.15 (i) If ko/n 2/ 3 _- o, then almost no Gp contains an isolated tree of order at least ko. (ii) Let 2 > 0 be a constant and let ci = cl(n) > 0 be such that el(n)n4/15 - oo and cl(n) -> 0. Furthermore, let c 2 (n) > cl(n) satisfy 2irc2 C1 (n) 5/ 2 .

el (n) -

c2(n) -

2 3

Then the number of isolated trees in G11 , whose orders are between ci(n)n / and c 2 (n)n 2 / 3 tends to the Poisson distribution P(2). (iii) If ko/n 2/ 3 -* 0, then a.e. Gl/n contains an isolated tree of order at least ko. Proof (i) Suppose ko/n

2 3

/

oo. Then by Theorem 5.13

-

n

Ep (ý-_

Tk)

•

k=ko

Eok/n(Tk) =O(n) E kk

k-5/2 = o(l).

k=ko

(ii) If k = o(n 3 / 4 ), then El/n(Tk)

=

(n) nk

kk-

51 2

2

n-k+l

_)

-l

k)-(n-k) (l

k{nl(k+3)/2}+1

-

)k(n

k/2)

5.3 The Largest Tree Components

3

nk -5/2 ,

-k(n-k/2) nk-5

(k

exp (n-k) ý,1-7r(

+

k2

n +2n

I

2n2) I

115 k3

2 + +3n 3

N

01}

12 3

exp{-k

/(6n 2 )}.

(5.28)

5 2 3 In particular, if k = o(n / ), then El/l(Tk) - nk- /// 2'. Hence with U = Y{Tk c1n2/ 3 < k •_ C2n2/ 3} we have

E(U) -

n•

S(c

2

k 5/2 .cln 2/ 3 < k
- cl)n2/

3

n (cl n2/3)-5/2

5/2z =

~,.

(c 2 - ci)cI 5/2/2i

In the second relation above we made use of the fact that {c 2 (n) cj(n)}n 2 / 3

--,

-

c0.

Analogously to part (iii) of the proof of Theorem 5.14, denote by E(Tk, Tk2, ... , Tk,) the expected number of ordered r-tuples of components of G1/, such that the ith component is a tree of order ki. If K

•

=

ki

=

o(n 2 / 3 ), then E(Tk, Tk2 .... ITkr)/E(Tkl)E(Tk2 )...E(Tk,) -

(n)

)

(k,k2,n..,kr)/{iiii

(k

(n+2 Ln/

(

E i=1

-X<

kkn

-Z(n-ki) -,+
K=Kn2

n-

-

:i)x<•kii}

k, k,

.)

n-K) nexp

S exp

(I

n

i=1n 2

- + Zkikj/n

2n

-

kikj i<j

1.

(5.29)

i<j

Consequently Er(U) -

{E(U)}r _ ,r

for every r > 1, and so our assertion follows from Theorem 1.21. (iii) This is an immediate consequence of (ii).

El

116

The Evolution of Random Graphs-Sparse Components

From the proof above it is easy to calculate the expected number of isolated trees of order at least cn 2/ 3 in G1/n, where c > 0 is a constant. Let V = E' Tk, where E' denotes summation over cn2/ 3 < k • n. Then E(V)

=

E

(E,

Tk)

_

n

1'k512explk3/6n')

where E denotes the expectation in §/,,. With the substitutions u and v = u 3 /6 we find that 3 1 fV /6uc) 1-5/_32e-ud E(V) -

,u5/2

e- U3/6 du

6

/)

13

-

-

k/n 2 / 3

y3/6

1.0 0.8 0.6 0.4 0.2 0.0 0.0

1.0

0.5

1.5

2.0

2.5

Fig. 5.3. The function p(c) = (1/6/F3n) f,3/ 6 v-3/ 2e-'dv, the limit of the expected number of isolated trees of order at least cn2 / 3 in G11 ,. It is tempting to conjecture [and it was claimed by Erd6s and R~nyi (1960, Theorem 7b)] that V tends to P(p). In fact, this is not so. If cn 21 3 <_ kl •_ k2 , then similarly to (5.29) we find that

E(Tk,,Tk2)/E(Tk,)E(Tk 2 )

kj n 2

(n) (n)

(

1

exp{-(kik2 + k2k 2 )/2n2} • e-'. Hence Ez(V) •< e-C 3{E(V)}2 = e c3•2, so V

d

P(2) does not hold.

k~k

2

5.4 Components Containing Cycles

117

5.4 Components Containing Cycles We saw in §2 that if p - c/n and c > 1, then a substantial proportion of the vertices of Gp lies on components containing cycles. Therefore it is not surprising that the study of the distribution of the components containing cycles is an important part of our investigation of the global structure of Gp for p - c/n, c > 1. The case of components which are cycles is rather simple. Theorem 5.16 Denote by Ck the number of those components in a r.g. Gp which are cycles of order k = k(n) >Ž 3. (i)

If pn

-*

0 or pn

-+

oo, then Ck = O Jor a.e. Gp.

(ii) If p c/n Jbr some constant c > 0 and k is constant, then Ck-P { (ce-c)k /2k}. (iii) If k = k(n) --* oo, then Ck = Ofor a.e. Gp.

d

Proof If p is sufficiently large, say p = 0(n)/n Ž! 3 log n/n, then a.e. Gp is connected (Ex. 14 of Chapter 2) so Ck(Gp) = 0 for a.e. Gp. Furthermore, if p = o(1/n), then almost no Gp contains a cycle (Ex. I of Chapter 4). Therefore one may assume that e _• O(n) _• 3 log n for some E > 0. Clearly ( k= (n)

(k - 1)! pk qk(fl-k)+(k)-k .

(n)k (O)k

(i

-O)kflk(k+3)/2

1 (n)k 0 k exp{-kO + k 2 0/2n + 3kO/2n}. 2k nk With a little work one can show that if either k(n) -* oo or O(n) -- oo then E(Ck) -* 0. Finally, if k is constant and p - c/n for some constant c > 0, then E(Ck) -

2 Cke-kc

=

and for every r > 1 Er(Ck) _ 2k, so Ck

-*

P;. by Theorem 1.20. The details are left to the reader (Ex. 5).

It is interesting to note that isolated cycles behave rather differently from isolated trees. By Theorem 5.16, the maximum of the expected number of components which are k-cycles increases up to p - 1/n and from then on it decreases. Hence about the same value of p gives the

118

The Evolution of Random Graphs-SparseComponents

maximum for every k. Furthermore, this maximum is less than 1 for every k. In fact, for p = 1/n the expected number of components which are cycles is 1 e k/ E( k=3

-- 1

e-k1k

Ck)

1

1 log(e - 1)

=

2

1

42

k=3

= 0.011564 .... The study of the components other than cycles needs some preparation. Denote by 16(n, m) the set of connected graphs with vertex set V = {1, 2,..., n} having m edges, and put C(n, m) = IW?(n, m)1. Clearly W6(n, n + k) = 0 if k < -2, W(n,n- 1) is the set of trees so C(n,n- 1) = nn-2, and 16(n, n) is the set of connected unicyclic graphs. The function C(n, n) was determined by Katz (1955) and R~nyi (1959b). The formula can also be read out of more general results of Ford and Uhlenbeck (1957), and another proof was given by Wright (1977a). Here we shall prove it from the following lemma, due to R~nyi (1959a) but asserted already by Cayley (1889). Lemma 5.17 Let 1 < k < n. Denote by F(n,k) the number of forests on V = {1,2,..., n} which have k components and in which the vertices 1, 2...,k belong to distinct components. Then

F(n,k) = kn"- -k

(5.30)

Proof The lemma can be proved in many ways. Here we shall present two proofs. Both are different from those given by Katz (1955), R~nyi (1959a) and Wright (1977a). Our first proof is based on the following special case of Abel's generalization of the binomial formula:

S(?

(x + l)- 1 (y + m

-

l)m-1-1 = (X 1 + y 1 )(x + y + m)m- 1 . (5.31)

1=0 k-

For (5.31) see Riordan (1968, p. 23) or Gould (1972, p. 15). Note that for k = 1 formula (5.30) is just Cayley's formula for the number of trees. Assume that (5.30) holds for k - 1, where k > 2. Then the number of appropriate forests with 1 + 1 vertices in the component containing vertex 1 is clearly

(n - k (l+ 1)'-'F(n-l- 1,k--1)

=

(n - k)

(ll)l_(kl)(nll)nk~ll.

5.4 Components Containing Cycles On putting x = 1,y

=

119

k - I and m = n - k into (5.31) we find that

n-k

F(n,k)

n /=0

k ) (I + 1)'-'(k

-

1)(n

-

I-

i

nk-1-

=

kn n-kl,

/

proving (5.30). Our second proof is based on Priifer codes for trees. The forests appearing in the lemma are in one-to-one correspondence with the trees on V' = {0, 1, 2,..., n} in which the neighbours of vertex 0 are 1, 2,..., k. If the code of a tree on V' is constructed by writing down the neighbour of the current end vertex with the maximal label then the code we obtain is such that the last k - 1 elements are O's, 0 does not occur anywhere else and the element preceding the O's is one of 1, 2,..., k. Conversely, every such code belongs to a tree in which the neighbours of 0 are 1,2, .... , k. Since the number of such codes is n'- -kk, our lemma follows. LJ

The formula for C(n, n) is an easy consequence of Lemma 5.17. Theorem 5.18 C(n,n)=

)

(

n'2r=3 j=

()

=

n=j

Proof A connected graph of order n and size n is unicyclic. What is the number of such graphs in which the unique cycle contains r edges? There are (n)r1= (n)r/(Zr)

(;)r,=

ways of choosing an r-cycle Cr. After the omission of the edges of Cr the graph becomes a forest with r components and the vertices of C' belong to distinct components. Hence C' is the unique cycle in F(n, r) of our graphs. Therefore

( n) C(n, n)

(

I

F(n, r) =

r=3 Ilnn

)E

(•r/8) 1/ 2nn 1/ 2 .

-(nn• r -i

1 (n)rn r=3

-I

r=3 j=1

Corollary 5.19 C(n,n)

I -

120

The Evolution of Random Graphs Sparse Components

Proof Since (I - j)

-Lexp{-()

n}

j=1

and for r = 0(n 21 3) (1-I

1)

= e-

2

0(r/n) + O(r3/n2)j,

/(2n){I +

j=l

we have

n

1 -/

)(1)

e-r22n) +

r=3 j=1

r=3

n

-

j

eX 2/2dx + 0(1) = (2

0(1).

Therefore C(n, n)

=

(71/8) 1/ 2 nn-l/ 2 + O(n- 1 ).

El

The use of Lemma 5.17 enabled R~nyi (1959b) to determine the asymptotic distribution of the length of the unique cycle contained in a connected graph of order n and size n (Ex. 7). We shall present a number of consequences of Theorem 5.18 concerning the unicyclic components of random graphs, but first we continue the examination of the function C(n, n + k). Bagaev (1973) showed that C(n, n+1) _- 54n+, and for 2 < k = o(n 1/3) Wright (1977a, 1978b, 1980, 1983) determined the asymptotic value of C(n,n+k): C(n,n + k)

ykn

{3k-1)/2{1 + O(k3/ 2n 1/2)},

+

where 7l/

Yk

2

3k(k -

I)Ak

2(5k_1)/2F(k/2)

and 5 1 =2

In fact, if k

=

C(n, n + k)

=

=

o(n1/ 3) and k 3)1/2

I 68h~k-h

k

36 ,k+1

6k + E h=1

k

k-

k

>

2.

oo then

-+

k

n(

3

k)

2

+ O(k3/ 2 n- 1 / 2 ) + O(k-)

5.4 Components Containing Cycles

121

Additional numerical data were given by Gray, Murray and Young (1977). The results above were considerably extended by Luczak (1990a) who showed that the last formula is a good approximation of C(n, n + k) provided k = o(n) and k --* oo. Furthermore, Bender, Canfield and McKay

(1990) proved a rather complicated asymptotic formula for C(n, n + k) for every function k = k(n) as n --* (Y.

In order to study the evolution of random graphs we shall not need an asymptotic formula for C(n, n + k), but we do need the following upper bound, due to Bollobits (1984a). Theorem 5.20 There is an absolute constant c such that for 1 < k < n C(n, n + k) •_ (c/k)k/2nn+(ak-1)12. Proof Recall that W(n, n+k) is the set of connected graphs with vertex set V = {1, 2... n} having n + k edges. Every graph GEcW(n, n + k) contains a unique maximal connected subgraph H in which every vertex has degree at least 2. Let W be the vertex set of H. Clearly e(H) = w + k, where w = IW1. Furthermore, G is obtained from H by adding to it a forest with vertex set W, in which each component contains exactly one vertex of W. Hence wnn-I-w graphs GeW(n, n + k) give the same graph H. Denote by T the set of branch vertices of H and by U = W - T the set of vertices of degree 2. Set t = ITI and u = IU = w - t. Clearly, H

is obtained from a multigraph M (i.e. a 'graph' which may contain loops and multiple edges) with vertex set T by subdividing the edges with the vertices from U (see Fig. 5.4). The multigraph M has t + k edges and loops, so there are at most u!B(u + t + k - 1, t + k - 1) ways of inserting the u vertices in order to obtain H from M. Since every vertex of M has degree at least 3 (a loop contributing 2 to the degree of the vertex incident with it), 3t_< 2(t +k),

that is, t < 2k.

Consequently 2k

C(n, n+ k) •S(n~) tp(t, t +k) tl

x E=

(n u-

+tt+ t +k- k -I 1) (t + u)nn÷l-tU, (5.32)

u=O

where ip(t, t +k) is the number of labelled multigraphs of order t in which

122

The Evolution of Random Graphs Sparse Components

2 2

14

U4

U5

Fi . .5 A g a h H 4

a mutg ' a h M wt

nd

TM

.

U6 ..

7

t

. .,

n

4

/3

us

9

U 10

Fig. 5.4. A graph H and a multigraph M with T

=tl

{t1

t4}

and U

every vertex has degree at least 3. Rather crudely

ip(t, t+ k) •! B

{(2

+t +(t + k- 1),t + k- 1(5.33)

since a multigraph M is determined by the partition of its t + k edges and loops into ( ) + t classes, (t) classes for (multiple) edges and t classes for (multiple) loops. To prove the theorem we shall simply estimate the bound for C(n, n+k) implied by (5.32) and (5.33). We shall write c 1,c2 .... for positive absolute constants. Note first that if t < 2k, then (t 2/2 +t +3t/2 k - +k-l) I

-(l~~-

nn

n

and (

t u!

=

t!+.

exp(--u2 /2n).

Therefore 2k

n-t

C(n, n + k) <_nn'-

A(t, u), t=I u=O

5.4 Components Containing Cycles

123

where A(t,u)

(lk

exp(--u 2 /2n)

)k't-

u+ t + k-

1

(t + u).

Let us split the double sum above into two parts, according to the size of u. Clearly 2k

2k

k

S S A(t, u)

k

t1(c, ,kttk exp(_u 2 /2n)2t+k(3k)

<

t=1 u=l

t=l u=i 2k 1 nn /2

5

(C 2 k)t-k

• nl/2(Ok)k-312

l/t!

t=1

<(c

3

(5.34)

/k)k/2n R12-1.

The case in which u is large is also easily dealt with: 2k

2k

n-t

5 5

A(t,u)

(c 4 k)t k1exp(

t

t=Iu=k+l

ut+k

exp(--U2

n-t

•!

/2n)

(t•-1)"

=

Since by Ex. 8 of Chapter 1 of xt+e-x2 /'dx <_ {(t + k)!}/ 2 , 2

5

A(t,u) <

t=l u=k+l

{(t +k)!}'/2 {(csk)t+k-I

n(t+k+1)/2

t=l

~t 52kný~)2

<

(k'/~ n<~

<

n,(k+l)/2 E(cAk)t/2+k/2nt/2/t!

<

n(k+l)/2 (c8/k)k/

2

12 k)!}U (t(c +6 k)t•k

2

nt1

2k

t=1 2

nk.

(5.35)

The assertion of our theorem follows from inequalities (5.34) and (5.35):

C(n, n + k)

nn- 1 {(c 3 /k)k/ 2 n3 k/2<_(elk)k12 n+(3k-1)/2.

<_

+- (c8±k)k12n(3k+1)/2}

If k is close to n then the bound in the theorem above is worse than the bound, valid for all k _>-1, given by the total number of graphs of

124

The Evolution of Random Graphs-Sparse Components

order n and size n + k: "

+k) 2(nen2)+k

+ k)< C~n~+ k <_n N Cn,n+k)

(5.36)

Inequality (5.36) and Theorem 5.20 have the following immediate consequence. Corollary 5.21 There is an absolute constant c such that C(n, n + k)

'for all n, k GN with k < N -n

=

3

< ck-k/2nn+( k l)/

(n)

2

n.

-

Proof Theorem 5.20 implies that in proving this corollary we may assume that n < k < N - n; in particular, n > 6. By inequality (5.36), in this range we have C(n, n+k) • < <

2

n k)n41k/ ( nr(e)n-+k ( n 2 exp n k+

In

e2)n+kexp { (

e

k(n + k/2)

k )k/ _

2

nk/2

kk/

}

2

3 nn+( kl)/

2

3

2

k1k/ nn+( k-l)/

2

e -(n+k)/2 k-k/2 nn+(ak-1)/2 nk 2

/(k12 n+(3k-1)/ ,


n

as claimed. With a little more work one can show that c = 1 will do in Theorem 5.20. Using this sharpening of Theorem 5.20, the proof above shows that c = 1 will do in Corollary 5.21 as well. As mentioned earlier, Theorem 5.18 and Corollary 5.19 can be used to obtain fairly precise information about the unicyclic components of random graphs. Our first result concerns components of bounded order. Theorem 5.22 Let p ,- c/n,0 < c < ao and denote by Xk = Xk(Gp) the number of unicyclic components of order k in Gp. Let 3 •_ k 1 < k2 < ... < ks be fixed. Then Xk1 , Xk ...... Xk, are asymptotically independent Poisson random variables with means 1 2 .. s, where k,-3

A,= 2(ce- )kE Zk/jl!. j=o

125

5.4 Components Containing Cycles Proof The joint factorial moments of mated:

C(ksk)()

rl

E{(Xk)ri "(1

X,k...,are easily esti-

Xk,,Xk

(I

-

1-I[(ce-")k'ri{C (ki,ki)/ki!}'r'/n !], i=1

where k

=1

=E

riki. By Theorem 5.18 this expression is exactly S

Hence Theorem 1.21 implies the result. Theorem 5.23 Denote by X = X(Gp) the number of vertices of Gp belonging to unicyclic components. Then for p

c/n, 0 < c < oo, c = 1, we have

-

k--3

ýo

E (X)

E 2(ce_,)k EkJ/j!

Y k(ce_

cc,

k-3 E Zki/j!

k=3

and for p

=

= y(c),

j=0

k=3

j=O

1/n we have 0.406n 2/ 3

E(X) - 6'/ 3LF(1/3)n2/3 12 and a2(X) =

Proof (i) Suppose p E(X)

=

"-

c/n,O < c < oc, c * 1. Then by Theorem 5.18 k--3k k

5kE(Xk) k=3

C(kk)pkqk(n-k)+( )k

k=3 n

~

(n)k k

k=3

2_1

O(n 4 13 ).

k=3

2

c~kflk /2-3k/2k-

(ce

,)k j=O

)=0i/!

kJ/j!

126

The Evolution of Random Graphs Sparse Components 10 8 6 4 2 0

12

3

4

3

j 4

25 20 15 105 0 0

1

2

Fig. 5.5. The curves of E ,/(X) and a,'/n(X). 1

Furthermore, as in the proof of Theorem 5.22, for fixed k, and k2 we have E{(Xký)2} - E(Xk,) 2 ,E{(Xk,) 2 } - E(X,)

2

+ E(Xkl),

and E(XkXk2 )

-"

E(Xk.)E(Xk2).

Consequently, for every fixed ko > 3: (E (kxk) kXk \k=3

2

/

k' )2ko+I - EE ( YkXk k=3

/

k2E(Xk). k=3

From this it follows that 2

0"2(X)

Zk k=3

,k-3 2

E(Xk) -

k(ce-c)k EkJ/.! k=3

since the last double sum is convergent.

j=0

5.4 Components Containing Cycles (ii) Suppose now that p

=

1/n. Then by Corollary 5.19

k(n~) g1/2 kk-1/2n -k (I

E (X)

n

I

-,4 ý_.

127

2

kn-k

k (n)k

-

I) kflk12-3k/2

/2-3k/2

n-

nk

k=3

n41 2

k-l_I (i

2 ek /2n+3k/2n

" k=3

-

j=1

Since k-(

1 -) H-n

<exp{

k(k-1) 2n

k(k-1)(2k-1) 6n2'

k4/(4n3)}

j=1

in the sum above we may assume that k = o(n 3/ 4 ). Now if k =o(n3/4), then

I-(

k2

k-_

- j)

=exp{k

k

_

2

6n2 k +0(

k4 3 )

j=l

}n

Consequently

E(X)

k exp os

x- 2/ 3 eXdx_ 6'/ 3 IF(1/3)n2/3"

61/3 2/3 12

4

Vn

J

12

The bound on the variance follows by straightforward manipulations.

The theorem above implies a more precise estimate of the number of vertices on unicyclic components than the one provided by Corollary 5.8. Corollary 5.24 (i) Given 0 < c < oo, c * 1, there is a b = b(c) such that

P{X(Gc/n)

> t}

<• b/t

In particular,if wo(n) ---* cc, then a.e. Gc/1 unicyclic components.

2

for all t > 0. has at most co(n) vertices on its

2 3 (ii) If wo(n) -* cc then a.e. G11 , has at most (o(n)n / vertices on its unicyclic components.

128

The Evolution of Random Graphs Sparse Components

Proof Both assertions follow from Theorem 5.23 and Chebyshev's inequality D

Exercises 5.1

Prove that if k > 2 is fixed and p = p(n) = o(n- 1), then cl(n, k,p)+ c 2(n,k,p) = o(1), where cl and C2 are as in Theorem 5.1. Deduce that dj{Y-(Tk),PJ --* 0,

5.2 5.3

where i = E(Tk) is the expected number of tree components of Gp having order k. Prove Corollary 5.2. Use Chebyshev's inequality to prove the following assertion concerning the tree components of G11,: if e > 0 is a constant and the functions r = r(n),s = s(n) satisfy 1 < r :< s < n, then lir•PI(n

kTk-n(

ý

kn , e

> gn

=0

kk=r

and lim PI/,

Tk -

ns kk

ek

=0.

k=r

5.4

By classifying the forests according to the number of neighbours joined to k specified vertices, deduce that "n-k t) k t F(n - kt).

" (n

F(nk) =

t~ l

5.5

Use this to prove Lemma 5.17 by induction on n. [Gdbel (1963), see also Moon (1970a, p. 17).] By considering rooted forests with n vertices and k components, show that for 1 < k < n - 1 we have

( n) knn

kj,

5.6

-'-l =

(

njn

ni, n2,

nm

n l-1...n , m-1,

where the summation is over all ordered partitions (ni, n2, ... of n - j. (Katz, 1955.) Prove that there are

(n

(n--1)n-k 1

,nm)

Exercises

5.7

129

trees on {1,2, ... ,n} in which vertex 1 has degree k. [Clarke (1958), see also R~nyi (1959a, p. 83) and Moon (1970a, p. 14).] Denote by y, the length of the unique cycle in a connected graph of order n and size n. Let x > 0 be fixed and set ro = [xn 1/2]. Show that, analogously to Theorem 5.16,

{r r-_ Y :!;fJ/f-j P(T} roJ

(1 -fi/j/n)

P(y, < r0) =

/

--+ 2F(x)-l.

r=3j=1

r=3 j=1

Deduce that the distribution of Wy/n1 /2 tends to the distribution of the absolute value of an N(0,1) r.v., so, in particular, E(7,) (2n/7r)1/2. (R~nyi, 1959b.) 5.8

The following proof of the first (and easy) part of Theorem 5.7(ii) was suggested by B. Pittel. Note that if the component of a vertex x, contains a cycle then the graph has a path xIx2 ... Xk and an edge XkXj for some

j, 1 < j < k - 2. Deduce that if 0 < c < 1 then the expected number of vertices of G/,l on components containing cycles is at most

(nk(kk=3

2 )
(I k=3

1k

- C)2-

6 The Evolution of Random Graphs-the Giant Component

The results of the previous chapter imply that for c < 1 a.e. graph process G= (G,)N is such that for t -,' lcn it satisfies Li(Gt) = O(logn), i.e. the maximum of the orders of the components of Gt is O(log n). As t passes n/ 2 , LI(G,) suddenly begins to grow. The main aim of this chapter is to investigate the speed of this growth. Erd6s and R~nyi (1960, 1961a) proved that a.e. G is such that LI(Gtn/21) has order n2/ 3 and if c > 1 then a.e. G satisfies L I(G[,n/ 2 ])

-

I -

t(c)}n.

Furthermore, a.e. G is such that GLcn/21 has a 'giant' component, that is a component whose order is much larger than the order of any other component. In this chapter we shall present the results of Bollobas (1984b), which shed considerably more light on the emergence of the giant component. If t is not much larger than n/2, then L1 (Gt) - 4t - 2n for a.e. G, i.e. the largest component grows about four times as fast as the number of edges. As the process continues, the larger components are swallowed up by the giant component at such a rate that the order of the second largest component decreases. By time cn/2, c > 1, the second largest component has order O(log n) for a.e. G. The final section is devoted to a result of Frieze (1985a) concerning economical spanning trees in graphs with random weights on the edges, and a result of Bollobais and Simon (1993) about the probabilistic analysis of the Weighted Quick-Find algorithm. 6.1 A Gap in the Sequence of Components The key result in the proof of the sudden emergence of the giant component is that from shortly after time n/2 most graph processes never 130

6.1 A Gap in the Sequence of Components

131

have a component whose order is between n2/3 and n2 / 3 . In fact, the gap becomes larger as the number of edges increases. The bound t _ n2/3. In particular,a.e. graph process is such that, for to = [(n/2)+sol < t < n, the graph G, has no component whose order is between n2/ 3/2 and n2/ 3. Proof A component of a graph will be called a (k,d)-component if it has k vertices and k + d edges. The number of (k, d)-components will be denoted by X(k,d). Since in our theorem the cases s < 0 and s > 0 have similar proofs, we shall assume that s > 0 and so t = n/2 + s > n/2. Furthermore, as the components we are interested in have order at most n2/ 3 , throughout the proof we shall suppose that ko(s) < k < n2/ 3. Denote by E,(k) the expected number of components of order k in G,: Et(k) = Z[Et{X(k,d)} : -- 1 •_ d < k*], where k* = (k) -k. 2 3 Set ki(s) = min{[n 2/ 3J,3ko(s)}. Then ko(so) < ½n / , the functions ko(s),ki(s) are monotone decreasing in so •< s < n/2 and ko(s+1) < 2k1 (s). Therefore it suffices to show that a.e. graph process is such that for so < s < n/2 and t = n/2 + s, the graph G, has no component whose order is at least ko(s) and at most k, (s). Indeed, suppose G is such a graph process, t < n - 1 and Gt has no component whose order is between ko(s) and n2/3. Since kj(s + 1) < 2k 0 (s), the (t + 1)st edge of the graph process cannot merge two components of Gt with fewer than ko(s) vertices to form a component of order at least kl(s + 1) in G,+±. Therefore, as Gt+1 has no component whose order is between ko(s + 1) and ki(s + 1), Gt+j does not contain a component whose order is between ko(s + 1) and n2/ 3. Consequently our theorem follows if we show that n

ki (s)

S E(k) = o(1). S t=to k=ko(s) Set s = s(t)

=

t - n/2,e = E(t) = (2t/n) - 1 = 2s/n and p

(6.1) - p(t)

=

132

The Evolution of Random Graphs-the Giant Component 2

2t/n = (1 + E)/n. A slight variant of Theorem 2.2(iii) implies that in our range Et(k) •4nl/2Ep(k), where Ep(k)

,-[Ep{X(k,d)} :-1 •_ d •_ k*].

=

Hence instead of (6.1) it suffices to prove k,(s)

n

I

Ep(k) = o(n-1/2).

(6.2)

t=to k=ko(s)

By Corollary 5.21 we have C(k,k + d) •_ co 6

dkk+(3d-1)/2

for some absolute constant co, so (n)

Ep(k)

Ep{X(k,d)} •

=

co

k3d11 " -

k

6 k

d=--I

P

d=-I

x (1 - p)kn-k(k+3)/2 d < c1

(n)

k-2

< cznk-52exp

')k-

+ (-j+

ek 2

F3

{5•E2 1 ej-k cnC

•~~

}

(kn--)

kk-2pk- exp {-p

1+F (kn

-

.

k+ 2n

3

Thus Ep(k) <_ c3 nk

5 2

/ exp{-0.499

2

k +

3 ; k/3}

=Fp(k),

since k/n = o(s/n) = o(r). If p> 10-3, then

Fp(k + 1)/Fp(k) < 1

_ 10-7.

Therefore (6.2) follows if we prove that n

tl

E Fp{ko(s)} + E t=tj

ki (s)

1

t~to k=ko(s)

Fp(k) = o(n-1/2),

(6.3)

6.1 A Gap in the Sequence of Components where t1 = [(1.001/2)nJ. Now for tr < t < n, s = s(t) and k

133 =

ko(s) we

have Fp(k) <• c4nk-5/2 exp

3.9-

-

8n

logn

This shows that Fp{kl (s)} =O(n1 -3"9+4/3+

•

1

) = o(n-1/2),

t=tI

since s <_n/2 for t < n. Also, if to < t _
5 2

exp{-0.49e k}.

Consequently tl

E

kt(s)

E

t1

Fp(k)

c3n(log n)5/ 2n

t=to k=ko(s)

5

t=to tj

c>n- 4 (log n)-5 1 2

ki(s)

exp{-0.49e 2k}

E•-5s

Ys

k=ko(s)

exp{--1.9(3 logs - log n)} n) s2(log

t=to tl

o(1)Z S-2

=

°(n-1/2)'

t~to

completing the proof of (6.3).

U

Let us call a component small if it has fewer than n2/ 3/2 vertices, and large if it has at least n2/ 3 vertices. By Theorem 6.1 a.e. graph process is such that for to _
pi = Ep(Xi)

< e = r(n) _
A},

134

The Evolution of Random Graphs the Giant Component

and k=

If

max{k :(k,d) e A}.

> _(1 + 6) 2 /n and kZ{B + (1 + g) 2 /n} a <2i+2Ui+a 2 (X,) n

and if-I

< g < -(1

{

•

n, then

+ (1 +- n

2

)

}

i+1

+ g) 2 /n, then 2

o (Xi)

<

1J2i.

Proof Analogously to (6.2) we have EP(Xi)=

k(n) C(k,k + d)pk2d(1--p)k•-k)+(•)kd

(6.4)

(k,d)cA

Also, if (k1 , d1 ) and (k 2 , d 2) are distinct elements of A, then

EP{X(k 1 , dl)X(k 2, d2)} =( n) (n -kkl C(k1 , k1 + dl)C(k 2,k 2 + d 2 ) × k,+k2+dl+d2(1

__p)(kl+k,)(n-k-k k)+ ( k,+k2 )k

Ik,-k2 -d, -d2

(nkk +k2

= Ep{X(k1 ,d1 )}Ep{X(k2, d 2 )}

p(n-ki

(n)kn+k 2 (1Wk), (n4 2

.p)-kk2 (6.5)

Similarly, for (k, d) c A,

EI{X(k, d) 2}

= Ep{X(k, d)} + Ep{X(k, d)}2 (n)k(n)k (n)2k (1

_ p)k

2

(6.6)

Relations (6.5) and (6.6) give Ep(X7)

=

Ep(X 2 i) +

S {k'kiEp{X(ki'di)}Ep{X(k

(n)k,+k2 (1

2 'd 2 )}

p)-k~k2 "(kj, d), (k2, d2) E A

(n)k, (n)k2

It is easily seen (Ex. 1) that if 0 < x _ y < 1, then

I - y < (1 Consequently ((;)k7+k2

-

x) ex-y.

_n{k,kn.k n) h

(i-I

k2 + i)

~

n_

-!

. (6.7)

6.1 A Gap in the Sequence of Components

< exp {i( L

k 2 ±i)} =

exp(-k 1 k2 In).

135

(6.8)

Furthermore, by inequality (1.10) of Chapter 1,

+r >eXP

1-p=l-

-

l+E

(1

n

{

n

2C)2.

j

(6.9)

Inequalities (6.8) and (6.9) give (n)k,(+k2 (

exp

<

p)-klk,

klkn

+(l+)2klk2 n2

+2(l+e)k1k2

n

(n)k (n)k 2

2

e+

= expkk2

)

=) 1 +6(ki,k

2 ).

Now if E+ (1 + E)2 /n < 0, then 3(kl,k 2 ) < 0 and so, by (6.7),

ao2(Xi) _<ýEp(X2i) -= l2i. Finally, if r > -(1

+

E)

2

/n and k2{8 + (1 + E)2 /n} < n, then

8+(

2klk -- 2 6 (k1, k2) _<ý ik

( 1(±-)2

n

n Therefore (6.7) gives a

Ep(X2i) +

2(1+

2 Sn2

x Ep{X(k 2 ,d 2 )} S/t2i

(ki,dl),(k 2 ,d 2 ) E A}

2E + 2(1+r)2)2

+

n

n2

1 1 ) Z{k+ 1 k2+ Ep{X(kl,dj)}

)

i+1.[

Given a graph G of order n, denote by S(G) the maximum of {k :k < n 2/ 3 and G has a component of order k} U {0}. If p is not too close to the critical value l/n, then it is fairly easy to estimate S(Gp) for a random graph GP. Theorem 6.3 Let -1/4 define 5

< e = e(n) < 1/4,

8EJ>

2n- 1 / 3, p = (1 + E)/n and

g,(k) = log n - 5 logk + k{log(1 + 8) - e} + 2 log(l/E) + k 2 /2n. Let ko = ko(n) and k 2 = k 2 (n) < n 2 / 3 be such that g,(ko) -

oo

and

g,(k 2 ) --+ -00.

136

The Evolution of' Random Graphs-the Giant Component

Then a.e. Gp is such that S(Gp) < k 2

and ifn"g1 3 (log n)-

2

then a.e. Gp is such that

__+c,

ko < S(Gp). 2 Proof (i) As in the proof of Theorem 6.1, for k <_ n /3 k*

Y. Ep{X(k,d)}

• clEp{X(k,-l)}.

d=-1

Hence the expected number of components of Gp with order between k2 and n2/ 3 is at most n2/3

c, Z

Ep{X(k,-1)}

k=k 2


k2/2n)(1+

k 512 exp(k

-

k-5/2exp

k - k/2n + klog(l + E) -k(l

)k

1

1±

k=k

2

<_c2n

+•+

-•(I + 8)

n2/3

cn C3

E k -5/2 exp[kllog(1 + e) - e} + k2 E/2n] k=k2

2 < c4 nk25/ exp[k 2 {log(1 + e) -

C4 C

e} + k 2 v/2n]8-2

exp{gE(k 2 )}.

By our choice of k 2 we have g•(k 2) -• -oc, so almost no GP has a component whose order is between k 2 and n 2 / 3 . (ii) In the proof of the second inequality we may and shall assume that n r 3 (logn) 2 -* cc and k 0 -> cc. Set k, = [8(logn) .], A = {(k,-1) k0 < k _< k•} and let Xo be as in Theorem 6.2. Then g,(k,) -+ -cc and k,

= E(Xo) =o

=

L k=ko

k,;

E(Tk) > c5 n E k-5/ 2 exp[k{log(1 + e) - rf}. k=ko

Clearly k, > ko+v- 2 , so by the choice of ko we have/to

we may suppose that po does not grow too fast, sayPo Then

occ. Furthermore, =

,uoEk./n = O{poE(logn)r. 4n-} = o(1).

o{nj, 3 (log n)- 2}.

6.1 A Gap in the Sequence of Components

137

Theorem 6.2 implies that S 2(Xo)

- yo + 38P2/n

po0 + 3Pgk2In= 0o{1 + o(1)},

so by Chebyshev's inequality P(X > 0) -> 1.

11

Let us state some explicit bounds for S(Gp) implied by Theorem 6.3. Corollary 6.4 Let p = (1 + E)/n. (i) If 0 < e < 1/2, then a.e. GP satisfies S(Gp) •! 3(logn)e- 2 . (ii) If n-' 0 < e = o{(logn)-l}, 0 < 7O < 1/3, and wo(n) -- o, then IS(Gp) - (2 log n + 6 loge - 5 log log n)e-2[ < wo(n)g- 2 for a.e. Gp. Proof (i) As we may assume that (log n)e- 2 < n 2 / 3, all we have to check is that g,(3(logn)e 2 ) -* -c0. (ii) Straightforward calculations show that

k 0 = {2 log n + 6 log e - 5 log log n - wo(n)}e-2 and k2 =

{2 log n + 6 log e - 5 log log n + wo(n)}e-2

satisfy the conditions in Theorem 6.3.

El

The corollary above enables us to prove an analogue of Theorem 6.1 for the time after n. Theorem 6.5 A.e. graph process G = (G,)' is such that for t >Ž 5n/8 the graph G, has no small component of order at least 100 log n. Proof By Corollary 6.4, a.e. graph process is such that for some t satisfying 3n/5 < t < 5n/8 the graph G, has no small component whose order is at least 75 log n. Since the union of two components of order at most a = [100 log n] has order at most 2a < n2/ 3 , the assertion of the theorem will follow if we show that a.e. graph process is such that for t > 3n/5 the graph G, has no component whose order is at least a and at most 2a. Furthermore, since for t >_ 2nlog n a.e. Gt is connected (cf. Ex. 15 of Chapter 2), it suffices to prove this for t < 2n log n. Let 3n < t < 2nlogn and set c = 2t/n. Then the expected number of components of Gt having order at least a and at most 2a is

2a

k'fX(kd))

kk=a d=-1

.

2a ~(n) k*C(k, __a

k+d)

n2 ) -k

(t)

(Nl

138

The Evolution of Random Graphs-the Giant Component <_ co E(en kk-(/

2a

•ý

2

/

(f t-k+lIk))-1T

•O2a k=a

(-n5/2 ,N)k

c1 jien)'k

N

(N

kn)k

N

k=a 2a <

c2 n Y

2 ekk-5/ ck e-ck

k=a 2a =

c2 nEk-5

12

(c el-c)k,

k=a

where co, cl and c 2 are absolute constants. Since for c = 1.2 we have c - 1 - logc > 0.0176, the last expression is at most c 3 n(log n)-3/2n-1. 7 = o(n-1/2).

L

6.2 The Emergence of the Giant Component Given a graph process G = (G,)•', denote by wt the number of components of Gt and write ,t for the partition of V into the vertex sets of the components of G,. Note that for every t either = •+1 6 or else wt+1 = Wt - 1 and £?'t is a refinement of Pt+l. Recall that a component is small if it has fewer than n2/3/2 vertices, and it is large if it has at least n2/3 vertices. Let G be a graph process which is such that for every t > to every component of G, is either small or large. Then for t >_ to the graph G,+l has at most as many large components as Gt, since no large component of Gt+T can be the union of two small components of Gt. Hence if t Ž> to and the graph Gt has a component which contains all large components of G,,, then Gt has a unique large component and so has every Gt, for t' > t. By Theorems 6.1 and 6.5 we know that a.e. graph process G satisfies the conditions above with to = [n/2 + (2 log n) 1 / 2 n2 /3]. Hence in order to prove that a.e. G is such that for t > t1 > to the graph Gt has a unique large component, it suffices to show that for a.e. G all large components of G,0 are contained in a single component of Gt,. Furthermore, the order of the giant component is the number of vertices which do not belong to the small components. Thus we shall know the approximate order of the giant component if we manage to estimate fairly precisely the number of vertices belonging to the small components of a typical G,. Since such an

6.2 The Emergence of the Giant Component

139

estimate will help us to prove that the large components of Gt0 are likely to merge soon after time to, we start with this estimate. Theorem 6.6 Let (log n) 1/Zn- 1/ 3 < • = o(1), p = (1 + e)/n and wo(n) --* co. Define 0 < e' < 1 by (1-r') e"' = (1+E) e-'. Denote by Y 1 (Gp) the number of vertices of Gp on the small (k, d)-components with d > 1, by Yo(Gp) the number of vertices on the small unicyclic components and by Y-I(Gp) the number of vertices on the small tree components. Then a.e. Gp is such that (i) Y1 = 0, (ii) Yo - o(n)- 2 ,

(iii) Yý1

-

n nn

(n)v8

Y-

n + 2En[ • w(n)e-/ 2 n1/

and 2

+ O(82n).

Proof As in the proof of Theorem 6.3, set k, = [8(log n)E-2]. Since g,(k,) -+ -oc, by Theorem 6.3 almost no Gp has a small component of order at least k,. Hence we may replace the Y1's by k,:

k,

S

)= >

kX(k, d),

k,)

k=4 d>1

kx(k,0)

= k=3

and E= kX(k,-1). k=1

(i) Corollary 5.21 and k, = 0(n2 1 3 ) imply that for k < k, E (ZX(k, d)) Hence, with

1

=

0(l)E{X(k, 1)} -_k

4

=

0(l) (n) kk± pk+I (1 P)kn-k2l2

Ed_>l X(k, d), we have

E(Yt) =

0(1) Z kkl

(1

k=4

0

k1/ex k~-)-

k -- 2n + Ik2n4

k, =

0(n- 1 ) L k-4

k012 eXp{-8 2 k/6},

8 - 2-+ O(83)

k -- (I + eOk-

(1--+ k 2n

140

The Evolution of Random Graphs the Giant Component

since ±3k k 2 /n = o(8 2k). This gives us E(Y 1*) = 0(n-' 8- 3 ) Therefore Y1 " = 0 almost surely, and so Y1

(ii) Recalling that C(k,k) ,,_ (7[/8)1/ tation of Yo is easily estimated: E(Yo)

2

o(1).

=

= 0 almost surely.

=f 2

kkk-/

(Corollary 5.19), the expec-

kE{X(k,0)}

= k _k,

=~~~

C kZI) (1

(n

V1 /7r/8 Z

)kl 1+ V

(

)k

k/2+3k/2

exp{-k. 2 /2 + k 2 /2n + O(kv 3) + 0(k/n)}

k k,

41 ) exp{-xv2 /2} dx=

el-2

Therefore, by Markov's inequality, P{0 -YoŽ(n)v7 2 }

0.

-2

(iii) It is easily checked that E' = e - le 2 + 0(v 3), since 0 < e' < 1 is

defined by (1 - F') e' = (1 + E)e-'. Hence it suffices to prove the first inequality. We shall estimate E(Y-I) rather precisely and then we shall make use of Theorem 6.2 to conclude that for a.e. Gp the variable Y-1 is rather close to its expectation. Set p' = (1 - e')/n and write E' for the expectation in <(n,p'). We shall exploit the fact that E and E' are rather closely related. First of all, calculations analogous to those in (i) and (ii) show that E'(YI) = E'(n - Yo - Y-l)

= 0(r7-)

and E'(Yo)

1 2 2

Hence

E'(

=

n - 1 2+ 2

o(E-

2

).

6.2 The Emergence of the Giant Component

141

In order to pass from E'(Y-) to E(Y_ 1), note that for k < k, we have E{X(k, -1)}/E'{X(k, -W)

i)t

)

1(

(1

+(1E )k-1

k~223/~~

1))

/(n

k(n 1)-k(k+1)/2+1

1-

E' + 8 1(n- 1) 1 exp{O(ek 2 /n)} 1+c 1+

+

(Ek 2 n). O

Consequently Sn-

1-2

O(l-2)

k 3 E'{X(k,-1)}

(/n)

}

k~l

-

1l - E'n 1cn

-1 - 2•+ O(E-2). 82

-

In fact, in order to apply Theorem 6.2, we need an even better estimate of an expectation similar to the one appearing above: k 2X(k,-1)

E

k-1

= O(n)

\k=l

2

exp(-kc2 /2)

k=l

= 0(n)

x-1/ 2 e

-,

/2 dx = 0(ne- 1).

2

Therefore, by Theorem 6.2 we have a-2(f_1)

O{n-

=

1

+ e(nl

2

1) /n}

=

0(n-').

Consequently, by Chebyshev's inequality a.e. Gp satisfies 11 If-, - E(Y- 1 )l = o~w(n)n

2 12 E'

Since E-2 = o(n1/ 29-1/ 2 ), this implies that -- T -1

1+

n/ < (o-(n)nl1/2E-1/2

n- •

for a.e. GP, as claimed. The property of having at most x vertices on the small tree components is monotone and so are the properties of having at least y vertices on the large components and at most z vertices on the small components which are trees or unicyclic graphs. Therefore by Theorem 2.2 we have the following consequence of Theorem 6.6.

142

The Evolution of Random Graphs the Giant Component

Corollary 6.7 Suppose t = (n/2) + s and '(logn)l/ 2 n2 /3 < s = o(n) and w(n) -* oo. Then a.e. G, has 4s + O(s 2/n) + O{wo(n)n/s1 /2 } vertices on its large components, at most wo(n)n 2/s 2 vertices on its small unicyclic components and n - 4s + O(s 2 /n) + O{wo(n)n/s1 / 2} vertices on its small treecomponents. Now we are ready to prove the emergence of the giant component shortly after time n/2. Theorem 6.8 A.e. graph process G = (Gt)u is such that for every t > tl [n/2 + 2(logn)l/ 2 n2/Sj the graph Gt has a unique component of order at least n2/ 3 and the other components have at most n2/ 3/2 vertices each. Proof Let so = (3logn)'1 2n2/3 . By Theorems 6.1 and 6.5, a.e. graph process is such that for t > to = [n/2 + so] every component of G, is either small or large. Therefore our theorem will follow if we show that a.e. G = (Gt)' is such that for some to < t • ti all large components of Gt, are contained in a single component of Gt,. By Corollary 6.7, a.e. Gto has at least 3,0 vertices on its large components. For such a graph H = Gto we can find disjoint subsets V1, V2. Vm of V(H) = V such that m > 5(log n)1/2 , IVII Ž n2/3,

i

1,...,m, M-

each Vi is contained in some V(Cj) and m

I

U

Vi = U

i=l

j=1

where C1, C 2, ... C1 are the large components of H = Gto. Set p = n-4/3 and denote by Hp the random graph obtained from H by adding to it edges independently and with probability p. Then a.e. Hp has at most to + n2/ 3 < t1 edges, so to prove our theorem it suffices to show that U'2 Vi is contained in a single component in a.e. Hp. What is the probability that for a given pair (ij), 1 < i < j • m, some edge of H1p joins a vertex Vi to a vertex of Vj? It is clearly at least ,

1

(l-

.p)4

I _ e-,

>

Hence the probability that in Hp all Vi's are contained in the same component is at least the probability that a random graph in W(m, 1/2)

6.3 Small Components after Time n/2

143

is connected. Since m -+ o, this probability tends to 1 (cf. Ex. 15 of

Chapter 2 and Theorem 7.3). This completes the proof of our theorem.

El

In the proof above p can be chosen to be n- 4/3(loglogn)(logn)- 1/2 . Then we obtain that the large components of Gto in a.e. G merge into a single component by time to + n2/ 3(log log n)(log n)- 1 / 2. Thus, once we have the gap in the sequence of the orders of the components and there are fairly many vertices on the large components, the giant component is formed in a very short time. Theorem 6.9 Let t = n/2 + s and wo(n) -> . If 2(log n)1 /2 n2 / 3 < s then a.e. Gt is such that

= o(n),

Li(G,) = 4s + O(s 2 /n) + O{wa(n)n/s1/ 2} and L 2 (Gt) •- (log n)n 2 /s 2 . If nl- '- < s = o(n/ log n) for some To < 1 then a.e. Gt satisfies 1L 2 (Gt) -

4(6 log s - 4 log n - log log n)n 2 /s 2 1 <_ o•(n)n 2 /s 2 .

Proof The assertion concerning LI(G,) follows from Theorem 6.8 and Corollary 6.7. The inequalities concerning L2(Gt) can be read out of Theorem 6.8 and Corollary 6.4, with e = 2s/n. Unfortunately, Corollary 6.4 cannot be applied as it stands, since the property in question is not convex, but the proof of Theorem 6.3 can be adapted to give an analogous El result for Gt instead of Gp (see Ex. 2).

6.3 Small Components after Time n/2 As t grows substantially larger than n/2, the global structure of a typical r.g. Gt becomes surprisingly simple: it contains no small components with many edges and all its small components have order O(log n). Theorem 6.10 Let 1 < co < cl be fixed and wo(n)

-*

o.

(i) Set zo = [con/2j, zl = [cln/2j and for r0 <_ t define c = c(t) = 2t/n, a(t) = c- I - logc and a = a(t) = [(1/ca)(logn - 2 loglogn +

co(n))J. Then a.e. G is such that for To - t < r1 the graph Gt has no small component of order at least a.

S=

144

The Evolution of Random Graphs-the Giant Component

(ii) A.e. G is such that if t > con/2, then every small component of G, is a unicyclic graph or a tree, and there are at most w(n)log n unicyclic components. (iii) For ow(n) -- oo a.e. G is such that if t Ž: wo(n)n, then every component of Gt, with the exception of its giant component, is a tree. Proof As most of the proof is rather similar to that of Theorem 6.3, we shall not give all the details. It is easily seen that if /# is sufficiently large, then for t > to the expected number of small components of Gt with at least b = [/f log nJ vertices is o(n-2). Hence a.e. G is such that if t > to, then Gt has no small component of order at least b. By Corollary 5.21, for t 0 < t = cn/2 < T2 = T1 • n and k •< b the expected number of components of order k in G, is at most

2E,{X(k,-1)i

(n )k k-2 ((n 2) (N;)

2

k

\t - k + I1/

2nkk -2N-kn+k k

2)t-k+l N,.k-2

2 2 2 • 2(en)kk-5/ e- kt/n(2t/n )k-1 = 4nk-5 1 2 (C el1-)k.

t N -- t + k-

)k-1 1

(6.10)

Once a small component appears in a r.g. process, it stays a component for a fairly long time. To be precise, for t < 2n log n and k = O(log n) the life-time of a component of order k of G, has approximately exponential distribution with mean n/2k (see Ex. 8). In particular, conditional on a component of order k, a(t) <_k •_ b, being present at time t, T0 < t < TI, it will remain a component of G,+,, Gt+2, ._ Gt+± with probability at least 1, where 1 = [n/4k]. Consequently, by (6.10) for k < b, P(for some To < t • ti the graph Gt has a component of order k) 4k 12 k -1/ 2 T2 _1(C ec)k, < Z 4nk 5 2 (c e, = l t•'t0

t=•-0

where c = 2t/n. Therefore the probability that for some to • t < T1 the graph Gt contains a component whose order is at least a = a(t) and at most b, is

6.3 Small Components after Time n/2

145

not more than 12

16

6

12

k -12(C

E

1-c

el-)k <

t=lo k=a(t)

16

a6- 3

1

c

a

t=r0 12

= O{(log n) 3/2} Z(c el-c)a t=T0

"12 = O{(log n)-3/21 E exp{- log n + 3 log log n - ow(n)}

=

o(1).

t=10

In the second inequality we made use of log(c el-c) = -a. This completes the proof of (i). The proofs of (ii) and (iii) are similar. 1The reason for the inequality t < cj in part (i) of the theorem was to enable us to use a fairly small function for a(t). With a' = [(2/a)log n] no upper bound has to be put on t (see Ex. 9). Let co > 1, c = 2t/n > co and let caand a be as in Theorem 6.10. By Corollary 5.19 the expected number of unicyclic components of Gt having order at most a is not more than =

()

k- 1

2

k

-(

2t

k

<

k=3

a-'(c

e -C)k=

0(1),

k=3

and the expected number of vertices on unicyclic components of G, is also 0(1). Recall that by Corollary 5.8 we have a similar assertion for c <1.

Theorem 5.9 gives a rather precise estimate of the number of vertices on tree components of Gp. This estimate can be carried over to our graph Gt, since the number of vertices on tree components is a monotone decreasing function: if G ( H and V(G) = V(H) then H has at most as many vertices on its tree components as G. Finally, an analogue of Theorem 5.10 gives an estimate of the maximum order of a tree component of Gt. Hence we arrive at the following result. Theorem 6.11 Let c > 1 be a constant, t = [cn/2J and ow(n)

->

oc. Then

a.e. Gt is the union of the giant component, the small unicyclic components and the small tree components. There are at most o)(n) vertices on the unicyclic components. The order of the giant component satisfies ILI (G,) -

{f1 - t(c)}nI <•

(n)n1/,

146

The Evolution of Random Graphs-the Giant Component

where (C) = 1 0 kk-I cE k k! (c e c)k, k=1

and Jor every fixed i > 2 L i (Gt)- (1/ce) where

logn-

5 loglognn

-o(n),

z = c - 1 - log c.

In the range of t covered by Theorem 6.9, the giant component increases about four times as fast as t. Another way of proving Theorem 6.9 would be to establish this fact first and then use it to deduce the assertion about the size of the giant component. What is the expectation of the increase of L,(Gt) as t = (1 + e)n/2 = n/2 + s changes to t + 1? The probability that the (t + 1)st edge will join the giant component to a component of order Lj is about L 1Lj/

().

Hence the expectation increase of Ll(Gt)

is about 2 (2L I/n ) :

k

2

(n) kk-

2

±

I

+1_

k

n

n

k=( n 2/3

2

I/n )E

•n k

(2)1/2 (Li/n) =(2

1/2

x-1/

(

S

/2exp(_E2 k/2) 2

e-x_2 /2 dx

(2/2)

1/2

= 2L 1 /(En)

=Li/s.

From this one can deduce that if t is o(n) but not too small, then L 1 (t) = Li(n/2 + s) - cis. Since t'(1) = -2, by Theorem 6.11 we must have c, = 4. In the rest of this section we shall investigate the number of components of G,. For t = [cn/2J, c < 1, Corollary 5.8 and Theorem 5.12 imply precise bounds on the number of components of almost every Gt. Theorem 6.11 shows that if c > 1 and t = [cn/2J, then most components of a.e. G, are trees. Since the number of tree components is a monotone decreasing function, Theorem 5.12 implies the following result.

6.3 Small Components after Time n/2

147

Theorem 6.12 Let 0 < c < x, c • 1, t = [cn/2j and denote by w = w(Gt)

the number of components of Gt. Then for wo(n) -+ c a.e. Gt satisfies ýw(Gt) - u(c)nl •_ co(n)n

where kk-2 c ec)k.

u(c) = 1

C Ek

If we sacrifice the approximation in Theorem 6.12, we can get an approximation of w(Gt) holding for all not too large values of t. Theorem 6.13 Let 2 < #3 < 1 be fixed and t = cn/2 where c = c(n) <

8 log n. Then P{fw(Gt)--u(c)ný Ž (logn)nf3 } < 100n 1- (log n)-. Proof Assume first that t [[n/2] + 1, [n/2] + 2, so that c • 1 or c > 1+ 4/n. Let p = c/n, k = [n n/ 2(logn)- 1 / 2], and denote by w*(G) the number of components of order at most k in a graph G. Put

A

{(k,

d) :1I < k :cýkI1!•d•:

(k)2~ }

and let X0, X 1 and yo, ul be as in Theorem 6.2. Note that our choice of the parameters ensures that the conditions of Theorem 6.2 are satisfied. By definition yo - Ep(Xo) = E(w*(Gp)). Furthermore, XI(Gp) is the number of vertices of Gp on components of order at most k so X, < n and It • n. Therefore, by Theorem 6.2, a-2 (Xo)

0

+-

n28(log n) 2

16logn+ n

+

2

n2J

• 17(log n)n. Hence, by Chebyshev's inequality P {jXo-pol ->!(logn)nfl}

_ 68(logn)-ln 1

2

.

(6.11)

Straightforward calculations imply that lpo - u(c)nl < n2/ 3.

(6.12)

2 w*(Gp) •! w(Gp) •E w*(Gp) + n1/ .

(6.13)

Clearly

148

The Evolution of Random Graphs-the Giant Component

Inequalities (6.11), (6.12) and (6.13) imply P {fw(Gp) - u(c)nl >_ 2(logn)nfi} _<68(logn)-f1 n1 2'l. Finally, since n > w(Gt) > w(Gt+l) Ž 1 for every t, by the De MoivreLaplace theorem (Theorem 1.6) we have for every t, 0 < t < 4nlogn, that P{Iw(Gt) - u(c)nI > (log n)nW} < 100(logn)-inl- 21f, as claimed.

[]

The function u(c) is monotone decreasing, and so is w(Gt) for every graph process G. Therefore Theorem 6.13 has the following immediate consequence. Corollary 6.14 The probability thatJbr a fixed #3a graph process satisfies

(Gt)o

Iw(Gt) - u(2t/n)nl <_2(log n)n'# Jor every t <_4nlogn is 1 - o(n2a3 fl).

El

6.4 Further Results The theorems in the previous sections are only the tip of the iceberg: Luczak and Wierman (1989), Luczak (1990b), Janson, Knuth, Luczak and Pittel (1993), Luczak, Pittel and Wierman (1994) and others proved numerous deep results about the behaviour of a random graph process (Gt) near its phase transition, i.e., near the time t = n/2. The deepest and most substantial study to date is that of Janson, Kauth, Luczak and Pittel (1993), whose approach was not probabilistic: they used precise estimates of generating functions to obtain remarkably detailed results. In this section we shall sketch only some of the important results about the phase transition and the supercriticalrange, i.e., the range beyond the phase transition. Let us define the excess exc(G) of a graph G as the maximal value of d for which G has a (k, d)-component, i.e., a component with k vertices and k + d edges. Call a connected graph complex if it contains at least two cycles, i.e., if its excess is at least 1. Using this terminology, Theorem 6.6.(i) states that in that range almost no GP is complex.

6.4 Further Results

149

Theorem 6.9 tells us that the giant component of Gt arises when t > n/2 + o)(n)n 2/3 and wo(n) _>2(log n)1/ 2. It is clear from the proof that this bound 2(logn)1/ 2 was only a slowly increasing function that enabled us to prove the required estimates. Luczak (1990b) showed that it suffices to assume that w)(n) -- o, so the 'window' of the phase transition of a random graph process is 0(n 2/3). Here we shall state this result of Luczak (1990b) for the entire supercritical range, but in a slightly weaker form. In Theorem 6.6 for E > 0 we defined 0 < e' < 1 by (1-e')e"' = (1+e)e-. Similarly, for s > 0, define 0 < s' < n/2 by ( n -

)

n2/)

Putting it slightly differently, with r = 2s/n, we set s' = e'n/2. Note that with p 1+8 and p' 1-8', a random graph Gp has about n + s edges and Gp, about

-2

s"

Theorem 6.15 Let t ILI(Gt)

-

=n

+ s, where s/n 2 / 3 - oo. Then a.e. Gt is such that

2(s+s')n = I L(Gt) -

'

n + 2s

E -

n2

5.

Furthermore,the largest component is complex, and every other component is either a tree or unicyclic, and has at most n2/ 3 vertices. Dl Luczak (1990b) gave the following precise estimate for the excess of Gt in the supercritical range. Theorem 6.16 Let o)(n) --* oo, and t

=

n/2 + s, where w)(n)n 2/ 3 < s <

n/co(n). Then a.e. Gt is such that exc(Gt)

-

16s3n

o-

0

s 3 /n 2.

U

Janson, Knuth, Luczak and Pittel (1993) extended this result: they showed that exc(Gt), suitably normalized, has asymptotically normal distribution. Taking up a suggestion of Erd6s, Janson (1987), Bollobfis (1988b) and Flajolet, Knuth and Pittel (1989) studied the length ýn = ý(G) of the first cycle that appears in a random graph process G = (G,)f' on n vertices. Rather interestingly, E(ý,) --* oo as n

distribution.

--

o,

but

,

has a beautiful limit

150

The Evolution of Random Graphs the Giant Component

Theorem 6.17 . For every k > 3, lim n-oo

P(ý,

=

k)

= 21 j' ) tkl et/2+t 2/4

1 _ tdt.

Furthermore, S"c/2 F(1/3) 21/632/3

6

In fact, Flajolet, Knuth and Pittel (1989) also accomplished the difficult task of giving precise estimates for all the moments of ý,, and Janson, Knuth, Luczak and Pittel (1993) also proved a number of related detailed results. The first cycle in a certain random directed graph process, thought to be relevant to the explanation of the origin of life based on the idea of self-organization, as expounded by Eigen and Schuster (1979), was studied by Bollobfis and Rasmussen (1989). Let us turn to the k-core of a graph, introduced by Bollobfis (1984a). For k > 2, the k-core corek(G) of a graph G is the maximal subgraph of minimal degree at least k. The 2-core is simply the core of the graph: core(G) = core 2 (G). As every graph of minimal degree at least 2, the core is made up of some vertex disjoint cycles and a set of branchvertices (vertices of degree at least 3) joined by vertex disjoint paths. There is a unique multigraph K of minimal degree at least 3 such that the 2-core of G is a subdivision of K. This multigraph K is the kernel ker(G) of G. Here the term 'multigraph' is used in a wide sense: in addition to loops and multiple edges, vertexless loops (called free loops) are allowed as well. Note that the multigraph M defined in the proof of Theorem 5.20 was precisely the kernel of G E W'(n, n + k). Luczak (1991a) conducted a detailed study of the core just beyond the critical probability n/2. Let us write Dk(G). For the number of vertices of degree k in the core of G. Among other results, Luczak (1991a) proved that if t = n/2 + s, where s = o(n) and s/n 2/ 3 -- o, then a.e. Gt is such that D2(Gt) 8s2 /n, D3(Gt) - 32s 3 /3n 2 and, for every fixed k, Dk(Gt) = O(sk/nkk-1) Furthermore, a.e. Gt contains an induced topological random cubic graph on about 32s 3/3n 2 vertices. Since by the theorem of Robinson and Wormald (1992) a.e. cubic graph has a Hamilton cycle (see Theorem 8.22), a.e. G, has a cycle that goes through about 32s 3/3n 2 cubic vertices of the core. By tightening and continuing this argument, Luczak (1991a) was able to give a good estimate of the circumference, the length of a longest cycle, of a random graph.

6.4 Further Results

151

Theorem 6.18 Let t = n/2 + s, where s = o(n) and s/n 2/ 3 -+ 00. Then a.e. Gt has circumference at least (16 + o(1))s2 /3n and at most 15s 2 /2n. El Luczak, Pittel and Wierman (1994) showed that the transition range from being planar to being non-planar is precisely the window of the phase-transition. Theorem 6.19 There is a function f : IR -* (0, 1) such that if c z R, t = t(n) = n/2 + s and s/n 2 / 3 _- c as n --+ oo, then lim IP(Gt is non-planar)= f(c). n--- o

Furthermore,lim,--,_

f(c) = 0 and lim,-oo f(c)

=

1.

The smallest non-planar subgraph of the first non-planar graph Gt in a random graph process G is likely to have order cn 21 3, so in some sense the problem of non-planarity is still a rather local question. This is not the case with the k-core for k > 3: the very first non-empty k-core is a 'giant' subgraph, with order proportional to n. To state this precisely, let -k be the hitting time of the k-core, i.e., for 6 rk(G) = mint :

Furthermore, let

yk(G)

=

corek(Gt)

(G,)N set

f 0}.

be the order of the k-core at the time it emerges: Ak(G) = corek(Gjk)I.

The following beautiful and difficult result about the sudden emergence of the k-core was proved by Pittel, Spencer and Wormald (1996). Theorem 6.20 For every k >_ 3 there are constants Ck > 1 and Pk > 0 such that Tk(G) - Ck n/ 2 and A (G) -'-pkn for a.e. graph process G. Furthermore, Ck and Pk can be expressed in terms of Poisson branching processes. In particular,Ck = min;,O0 Ork( ) = k + V/Li6gk + O(logk), where 7Ek() = P(Poisson(,) Ž k - 1). The above problem of finding a non-empty k-core looks rather similar to that of finding a subgraph that is not k-colourable. Nevertheless, the latter problem seems considerably harder and is rather far from being solved, as we shall see in §5 of Chapter 11. Molloy and Reed (1995) studied the phase transition in a refinement of the models S(n,p) and S(n, M). Let 2 0, 2 1, ... be a sequence of nonnegative reals with T = 1. Consider the space of all graphs on n labelled vertices in which the number ni of vertices of degree i satisfies

152

The Evolution of Random Graphs-the Giant Component

nj-2 1 nj < I for every i. Molloy and Reed (1995) proved that the existence of a giant component is governed by the quantity Q = E i(i - z)ýi. If Q > 0 then a.e. random graph in this space has a giant component, and if Q < 0 then almost no random graph has a giant component. In conclusion, let us say a few words about the various approaches to random graph problems, especially phase transitions. 'Statistical graph theory' had existed before Erd6s and R~nyi started their work on random graphs, but its aim was exact enumeration. Generating functions were found and then analytic methods were applied to obtain asymptotics. In this method there was nothing probabilistic, random variables and probability spaces were not mentioned. Erd6s and R~nyi introduced an entirely novel approach-rather than strining for exact enumeration, they adopted a probabilistic point of view and treated every graph property as a random variable on a certain probability space. This is precisely the method we apply much of the time in this book; in particular, the results in this chapter were obtained by this 'static' probabilistic method. It is fascinating to note that percolation theory (the theory of random subgraphs of lattices) was started by Broadbent and Hammersley (1957) at about the some time as Erd6s and R~nyi were laying the foundations of the theory of random graphs, but the two theories were developed with minimal interaction (see Grimmett (1999)). By now there is more and more interaction between the two disciplines; for example, the detailed study of the phase transition in the mean-field random-cluster model by BollobAs, Grimmett and Janson (1996) was based on the properties of the phase transition in the random graph process (Gt)N The original approach to random graphs based on generating functions is far from out of date; rather the opposite is true: it is more powerful than ever. This approach may require much work and technical expertise, but it can produce deep and difficult results. Perhaps the best examples of this can be found in the monumental paper of Janson, Knuth, Luczak and Pittel (1993) about the structure of a random graph process next to its phase transition. In addition to natural mixtures of these two basic approaches, there are several other lines of attack; perhaps the most exciting of these is the dynamical approach pioneered by Karp (1990). (The germs of this method can be found in the work of Erd6s and R6nyi.) In Karp's dynamical approach edges are 'uncovered' or 'tested' only when we need them: the random graph Gp unravels in front of our eyes, much as a birth process.

6.5 Two Applications

153

For example, to construct the vertex set of the component of a vertex u of Gp c W(n, p), we may construct a sequence (Ti, Ri)o of subsets of V(Gp) with Tic Ri as follows. Set To = 0 and R0 = {u}. Having constructed the random sets (Tk, Rk), if Tk = Rk then we set 1 = k and stop the sequence. Otherwise, pick a vertex tk+1 c l&/Tk, set Tk+I = Tk U {tk+l}, text the graph Gp for its potential edges joining tk+l to vertices outside Rk, and add all the neighbours of tk+± to /k to obtain Rk+1. What is happening is this: at time k set Tk is the set of vertices we have tested and R/ is the set of vertices we have reached from u. Clearly, T, = R, is precisely the vertex set of the component of u in Gp: at time 1 we have tested all the vertices that we have reached, so no other vertices can be reached from u. As ITkr I k for every k, the component of u has T1 = 1 vertices; furthermore, IRk+I\RkI has a binomial distribution with parameters n IRkI and p. To find the other components, we start the same process on the remainder V(Gp)\I T711,then on what remains after the removal of the next component, and so on. Needless to say, this is only the beginning, and much work needs to be done to obtain worthwhile results. Karp (1990) used this approach to describe precisely the component structure of a random directed graph near its phase transition, but the some approach can be used to prove sharp results about the component structure of random graphs. For some results in this vein, see BollobAs, Borgs, Chayes, Kim and Wilson (2000).

6.5 Two Applications There are several applications of the results concerning the global structure of random graph processes; here we shall describe two of them. Let ={Xi, 1: < i < j <_ n} be a sample of size N = (n) from the uniform distribution on [0, 1]. Assign weight Xij to the edge ij of the complete graph with vertex set V = {1,2,...,n} and denote by H, = H(X) the random weighted graph obtained in this way. Denote by s(Hn) the minimum weight of a spanning tree of H,. That is the expected value of s(Hn), and what is the probability that s(H,) is close to its expectation? Our first aim in this section is to present a theorem of Frieze (1985a) answering these questions. At the first sight it is rather surprising that the answers are very elegant. Earlier a slightly weaker bound on the expectation was obtained by Fenner and Frieze (1982). Related problems were studied by Lueker (1981), Steele (1981) and Walkup (1979); see

154

The Evolution of Random Graphs-the Giant Component

the discussion of the random assignment problem at the end of §30 Chapter 7. Theorem 6.21 (i) lim-

E{s(Hn)} =(3) =

-3 = 1.202 ....

(ii) For every E > 0 lim P{Is(Hn) -

(3)1 _ } = 0.

n---ID

Proof Since Hn is a function of X, we shall write s(X) for s(H,). Let X(j) _• X(2) _<... _< X(N) denote the order statistics of the sample X = (Xij). With probability 1 no two Xj's take the same value, so with probability 1 we have XAl) <X2) < ... < X(N). In this case X = (Xij) defines a graph process G = G(X) = (G,)' in a natural way: the edge set of G, is

{ij : Xij

= X(k)

for some k _ t}.

Thus the edges of G, are the t least expensive edges. For this graph process G set W(k) = mint "w(G,) = n-k},k = 1,2,...,n-

1.

Then Wp(I) = 1 and Wp(n - 1) is the first time t when the graph Gt is connected. By the greedy algorithm of Kruskal (1954) (see also Bollobis, 1979a, Chapter I) an economical spanning tree of Hn is formed by the 2 edges of G appearing at times p(l), Wp( ) ... , i(n - 1), i.e. n-1

S(X)

Xpt.

= k=l

By Ex. 15 in Chapter 2 P(G[2 nlognj

is disconnected) = O(n- 3 ),

(6.14)

so with probability 1 -O(n- 3 ) we have W(n- 1) _• 2nlog n. The r.v. X(,) is rather close to the constant i/(N + 1). To be precise, E(X(,)) = i/(N + 1) and by a standard result in probability theory for every e > 0 we have

sup {P{E ZXW-El/(N +1)

~

I a {1,2,..., [2nlognj} , 111

n-1} = o(1).

6.5 Two Applications

155

Therefore in our proof we may replace s(X) by n-1

s'(X)

+ 1).

= k=I

[n7 /8J. If io(n-

Set no = n-

1) < 2nlog n, then

(n(log n)nv7/s

?1--i

ZW(k)/(N +41)

0

)

=o()

=

0(1),

k=no

so we may replace s'(X) by no

s"(X)

=

j

ip(k)/(N + 1).

k=I

About how large is ip(k)? Since ip(k) is the first t for which Gt has n - k components, and by Corollary 6.14 a graph G, has about u(2t/n)n components, tp(k) is about Wpo(k), where Wpo(k) is defined by u{2To(k)/njn = n - k. To be precise, if k < no, then

u{2Wo(k)/n} Ž_ [n7/8j/ n and so 9 exp[-2{jpo(k) + n8/ }/n] > !n1/8

since u(c) e'

--

1 as c

oo. Easy calculations show that

-

lu'(c)l

>

e

(Ex. 10). Hence, for k < no u[2{ipo(k) + n8/ 9 }/n]n •_ n

-k

I n-1/82n8/9

4

< n - k - 3(log n)n3/ 4

(6.15)

+ 3(log n)n3 / 4.

(6.16)

and, similarly, u[2{ipo(k)

-

n8/ 9 }/n]n

By applying Corollary 6.14 with 11/4) 1w(Gt) -

>

1

n -k 3

=

4

we see that with probability

u(2t/n)nl <_ 2(log n)n3/ 4

(6.17)

for every t :s 4nlogn. Hence by (6.14), (6.15), (6.16) and (6.17) with probability 1 - o(n-1/4) Wo(k) - n 8/ 9 < lp(k) •_ ypo(k) + n8 / 9 .

(6.18)

156

The Evolution of Random Graphs the Giant Component

This leads us to our final approximation of s(X): no

so(n) = E wo(k)/(n + 1) k=I

Inequalities (6.14) and (6.18) show that E{s"(X)} = so(n) + O(n- 1'/ 9) and with probability 1 -- o(n-1/4) s"(X) = so(n) + 0(n-1/9). All that remains to show is that lim by straightforward calculations: lim so(n)

=

where to = po(no). Clearly to n - no

"• n 7/ 8 .

, so(n) = ý(3). This can be done -o(k)lim 2 / t t du(2t/n),

n limr2 = _=2 2 n

n-

0

k '-1'

)

n-a'o

n

•nlogn since

e- 2to/nn -

u(2to/n)n

-

Hence

lim so(n) = lim 2 n--onn

nc

=

t tu'(2t/n)

of xu'(x) dx =

n

dr = lim 42 f n-ýc, n Jo

nu(x)n dx 2 2

J) u(x) dx,

where the last equality followed by partial integration. Since 0kk

k-i1

-kx

k=1

we have kwkrn so(n)=Zo•kk!-x22 o xkk=

J

e-x dx= dX

s

=(3)

k=I

completing the proof. Steele (1987) noted that the above theorem of Frieze holds for a under class of distribution: if the distribution function F of Xij is differentiable from the right at 0 and a = F'(0) > 0 then s(H,) converges to ((3)/a in probability. This result was sharpened by Frieze and McDiarmid (1989): by making use of martingale inequalities like Theorems 1.19 and 1.20, they proved that s(H,) -- (3)/Lx with probability 1. The second application of random graph processes we shall present

157

6.5 Two Applications

concerns the probabilistic analysis of the Weighted Quick-Find (WQF) algorithm. First we have to set the scene. The Union-Find problem of Aho, Hopcroft and Ullman (1974) asks for maintaining a representation of a partition of an n-set S in such a way that the two self-explanatory operations called Find[x] and Union[x, y] can be carried out efficiently. One of the simplest algorithms proposed for this problem uses an array Name[x] to store the name of the equivalence class that contains x, and for each name s and linked list L[s] of all the elements in the equivalence class of u. The result of Find[x] for x G S is the name Name[x], and the result of Union[x,y] for x,y E S is a new partition of S in which the equivalence classes C, and Cy of x and y are merged into the single equivalence class Cx U Cy. To implement Union[x,y], where Name[x] : Name[y], one must set Name[u] <- Name[y] for each u in L[Name[x]] and append L[Name[x]] to L[Name[y]]. Hence the cost is proportional to CxJ, the cardinality of the equivalence class C, of x. By maintaining a separate array N[s] with the cardinality of the equivalence class s and changing the name of the elements in the smaller of the two classes, the cost Union[x,yJ can be reduced to min{ Cxl, Cy 1}. This is the Weighted Quick-Find algorithm. The average use behaviour was studied by Doyle and Rivert (1976), Yao (1976), Knuth and Sch6nhage (1978), and Bollob~is and Simon (1993) under various models; here we consider the model closest to random graphs. A random graph process G = (Gt)' has a sequence of Unions in WQF naturally associated with it: the tth operation is Union[x,y] if the tth edge in G or weight is xy, i.e. if Gt = Gt- 1 +xy. The cost or weight of this edge is w(x,y) = min{JC,, JCyJ} if xy joins two components of Gt_1, with vertex sets C, and CY. If xy joins two vertices of the some component of G,_1 then w(x, y) = 0. Finally, the cost W(G) of the graph process G relative to WQF is the sum of the costs of its edges. Here is then the result of Bollobfis and Simon (1993) concerning the expectation of W(G). Theorem 6.22 The expected cost of a random graph process G relative to the Weighted Quick-Find algorithm is E(G(W)) = con + O(n/ log n), where = log2-1+.I: k>1

= 2.0847 ....

X\

'

(k0l-2K I (k +

lk~I1

158

The Evolution of Random Graphs the Giant Component

Strictly speaking, the value 2.0847 ... for the BollobAs-Simon constant is due to Knuth (2000, Chapter 21), who gave precise estimates for it and determined it up to 45 decimal places. It is interesting to note, that the analogous expectation for the QuickFind algorithm in which w(xy) = (ICxj + ICyl)/2 is considerably larger, namely n2 /8 + O(n(log n) 2).

Exercises 6.1

Prove that if 0 _•x _•y •1, then

1- y < (1 - x) eX Y. 6.2

Suppose t = n/2 + o(n). Imitate the proof of Theorem 6.3 to show that a.e. G, satisfies S(Gt) <_(log n)n 2 /s 2. Show also that if n/2 + n1-7° < t = n/2 + s < n/2 + o{(logn)-ln} for some Yo < 1, and

wo(n) -+ o, then

2 2 2 2 IS(Gt) - (6 logs - 4logn - loglogn)n /s 1 <_wo(n)n /s

6.3

for a.e. Gt (cf. the proof of Theorem 9). Let1< < 1,E=n-,p=(1 +e)/n, and let Y1 = Y(Gp) be as in Theorem 6. Prove that there is a constant c', > 0 such that E(Y 1 ) = {c;. + o(1)}n 5 >-,

though Y1 6.4

=

0 almost surely.

Suppose To < 1, n-70 <- Ill = o{(logn)-l}, p = (1 + P)/n and

wo(n) --* o. Prove that for every fixed m > 2 a.e. Gp is such that if 2 < i_< m, then Li(Gp) - (2log n + 6log ýel - 5 log log n)P-21 < •o(n)r-2.

6.5

(This follows rather easily from Corollary 6.4 and Theorem 6.8.) Let c > 1. Prove that there is a constant cl such that max P(Gt contains a small component of order at least cI log n)

t>_cn/2

= o(n-2).

6.6

(Ex. 5 ctd.) Show for t > cn/2 that the expected number of (k, d)-components of Gt with k < cl logn and d > 1 is O(n- 1).

Exercises 6.7

6.8

6.9

6.10

159

(Ex. 5 ctd.) Prove that there is a constant c2 = c2(c, cI) such that for t > cn/2 the expected number of vertices of G, on unicyclic components of order at most c, log n is at most c 2. Prove that for t < 2nlogn and k = O(logn) the lifetime of a component of order k of Gt has approximately exponential distribution with mean n/(2k). For t > n/2 define c(t) = 2t/n, o#(t) = c - 1 - logc and a'(t) = [(2/o)logn]. Prove that for co > 1, a.e. graph process G is such that for t > con/2 the graph Gt has no small component of order at least a'(t) (cf. Theorem 6.10(i)). Show that the function u(c) of Theorem 6.12 satisfies Iu'(c)I > e-c (cf. the proof of Theorem 6.15).

7 Connectivity and Matchings

Perhaps the most basic property of a graph is that of being connected. Thus it is not surprising that the study of connectedness of a r.g. has a vast literature. In fact, for fear of upsetting the balance of the book we cannot attempt to give an account of all the results in the area. Appropriately, the very first random graph paper of Erd6s and R~nyi (1959) is devoted to the problem of connectedness, and so are two other of the earliest papers on r.gs: Gilbert (1959) and Austin et al. (1959). Erd6s and R~nyi proved that (n/2) log n is a sharp threshold function for connectedness. Gilbert gave recurrence formulae for the probability of connectedness of Gp (see Exx. 1 and 2). S. A. Stepanov (1969a, 1970a, b) and Kovalenko (1971) extended results of Erd6s and R~nyi to the model S{n,(pij)}, and Kelmans (1967a) extended the recurrence formulae of Gilbert. Other extensions are due to Ivchenko (1973b, 1975), Ivchenko and Medvedev (1973), Kordecki (1973) and Kovalenko (1975). In §1 we shall present some of these results in the context of the evolution of random graphs. We know from Chapter 6 that a.e. graph process is such that a giant component appears shortly after time n/2, and the number of vertices not on the giant component decreases exponentially. Eventually the giant component swallows up all other components and the graph becomes connected. Our aim in §1 is to examine the transition from a disconnected graph to a connected one. The edge-connectivity 2(Q) and the vertex-connectivity K(Gp) of a r.g. were examined by Erd6s and R6nyi (1961b), Ivchenko (1973b) and Bollobfis (1981a). Some of these results will be proved in §2. A complete (or perfect) matching in a graph of order n is a set of n/2 independent edges. It is trivial that if a graph is connected or if it has a complete matching then it has no isolated vertices. Erd6s and R&nyi 160

7.1 The Connectedness of Random Graphs

161

(1964, 1966) proved the following result, which is very surprising at first sight: if M is large enough to ensure that a.e. GM has minimum degree at least 1, then a.e. GM is connected and, if n is even, a.e. GM has a perfect matching. A sharp version of this result is due to Bollob~is and Thomason (1985): a.e. graph process is such that the moment the last isolated vertex vanishes, the graph becomes connected and, if n is even, the graph acquires a perfect matching. In §3 we consider matchings in bipartite graphs and in §4 we examine the general case. The aim of §5 is to take a quick look at the rich body of results concerning reliable networks or, what it amounts to, the probability of connectedness of G E S(H; p). Once again, we are faced with an impossible task. Rather than give an unsatisfactorily superficial account of good many results, we concentrate on a striking theorem of Margulis (1974a). The connectivity properties and 1-factors of random regular graphs will be examined in §6. 7.1 The Connectedness of Random Graphs Let us examine a graph process just before the giant component swallows up all the other components. We know from Theorem 6.10(iii) that if wo(n) ---> o, a.e. graph process is such that after time wo(n)n every component, other than the giant component, is a tree of order o(log n). )N When do these trees vanish? Given k ( N and a graph process G = set z(Tk)

= max{O} U {t : G, has a component which is a tree of order k}.

Thus T(Tk) is the last time our graph contains an isolated tree of order k, provided it ever contains one. As in Chapter 5, denote by Tk the number of components which are trees of order k. Clearly Ep( T = Ep(Tk) so if k = o(log n), k

--

( )k

~~n )) k-2pk-(

1 -p)kn+O(k

2

)'

oo, p = O/n and 0 = o{(n/k)1 / 2 }, then

Ep(Tk)" -

1 ,,2-,

k -5/2 n- (Oe

-Ok)

0

For k E N define 00 = Oo(k) > 1 by k 5/2n Oo- (ooe'-o)k =

1.

Connectivity and Matchings

162

Theorem 7.1 If k = o(logn) and wo(n) -- o, then Ir(Tk) - Oo(k)n/2 < o•)(n)n/k

holds for a.e. graph process. Proof (i) The lower bound on T(Tk) is rather trivial. Let tl = [{00 o)(n)/k}n/2] and 01 = 2t 1/n. Then

Et,(Tk) -. Eo,/n(Tk) -" ( n k)

k-_2

-

ee-kO,

0)

The definition of 00 implies that Et,(Tk) --* oo. Furthermore, rather trivially, ao(Tk) = o{Et,(Tk) 2} so Tk(Gt,) > 0 for a.e. Gt,. Hence r(Tk) > t1 for a.e. graph process. (ii) The proof of the upper bound is similar to the proof of Theorem 6.10. Since a.e. graph of size b = [2n log n] is connected (cf. Ex. 15 of Chapter 2), we may restrict our attention to times up to b. Furthermore, we may and shall assume that wo(n) <_loglog n. Set t2 = [{Oo + wo(n)/k}n/2J and 02 = 2t 2/n.

Then Et2 (Tk)

)

- E02/(Tk)

n( kk-

2

()

e-k0 2 = o(1).

Given a constant c > 0, set s = [cn/(2k)J and 0 = 2(t 2 + s)/n. It is easily seen that Et2,+(Tk) -' Eo/,,(Tk) < 2-cEo2/n(Tk).

Consequently 2b

Z

Et(Tk) = o(n/k).

(7.1)

t=t 2

The survival time of a tree component of order k is asymptotically exponentially distributed with mean n/(2k), so with probability 1 a tree component of order k in Gt, for t •< b will stay a component for at least n/(4k) time. But, as in the proof of Theorem 6.10, by relation (7.1) this implies that a.e. G = (G,)' is such that for t 2 •! t < b no Gt has a tree ED component of order k. As a graph process runs, the tree components are swallowed up by the giant component at a regular rate, those of higher order vanishing first.

7.1 The Connectedness of Random Graphs

163

Theorem 7.2 Suppose t/n -* oo and ko = ko(n) e N is such that E,(Tko) -* co and Et(Tko+1) -* 0. Then a.e. G, is such that all its components, with the exception of the giant component, are trees and {k : G, has a tree component of order k} = {1,2,...,ko}. Proof We know that a.e. Gt consists of a giant component and tree components of order o(log n). Hence in our proof it suffices to consider tree components of order at most logn. Furthermore, it is easily seen that for t >Ž 2nlogn we have maxEt(Tk0 ) -+ 0, so t < 2nlogn must hold. (i) Straightforward calculations show that for 1 k :5 log n Et( Tk+1 )/Et( Tk) - (2t/n)e-2 t/n. (7.2) Since t/n -

oo, this implies [lognj

Z

Et(Tk) = o(1),

k=ko+l

so almost no Gt has a tree component of order at least ko + 1. (ii) By Chebyshev's inequality the probability that there is a k, 1
E(Tk) 2 .

(7.3)

k=1 2

Now, for p = 2t/n the models ýV (n, t) and (n,p) are practically interchangeable. If we apply Theorem 6.2 to the one element family A = {(k,--1)}, then we see that UZ(Tk) •_ Ep(Tk) + 3tk 2 E(Tk) 2 /n 2 . Imitating the relevant calculations in the proof of Theorem 6.2 we find that a2(Tk) • E,(Tk) + 4tk2 Et(Tk)2 /n 2 . This shows that the sum appearing in (7.3) is at most k0

k0

Z 1/Et(Tk) + Z k=1

k=l

k0

4tk2/n

2

1/Et(Tk) + O(tk2/n 2 ).

= k=1

By relation (7.2) the sum on the right-hand side is o(1). Furthermore, 3 2 as k0 •_ logn and t _•nlogn, we have O(tk2/n ) = O{(logn) /n} = o(1).

164

Connectivity and Matchings

Recalling Theorem 5.4, we see that for a fixed k the trees of order k vanish when t > (n/2k){logn+ (k - 1) loglogn+to(n)}, where wo(n) -* oo. If t = (n/2k){logn + (k - 1)loglogn + x + o(l)} for some fixed x, then tree components of order k exist with probability I - e-e-1(kk!) + o(l). From Theorem 7.2 it is easy to deduce the fundamental result proved by Erd6s and R&nyi (1959) that the probability of a r.g. Gp being connected is about the probability that it has no vertex of degree 0. In view of the importance of this result we also give a proof which does not rely on the structure of a typical Gp. Theorem 7.3 Let c e IR be fixed and let M and p = {logn+c +o(l)}/n. Then P(GM is connected)

=

(n/2){logn + c + o(1)}

N

ee

(7.4)

e-e-.

(7.5)

•

and P(GP is connected)

Remark. By Theorem 2.2(ii), relations (7.4) and (7.5) are equivalent, so it suffices to prove one of them. 1st Proof By Theorem 7.2, a.e. GM consists of a giant component and isolated vertices. Hence P(GM is connected) = P{6(GM) >_ 1} + o(1).

By Theorems 3.5 and 2.2(ii), P 16(GM) > I} ---e->

'

In fact, recalling Theorem 5.4, we see also that the number of isolated vertices is asymptotically a Poisson r.v. with mean e-". 2nd Proof The probability that Gp has a component of order 2 (i.e. an isolated edge) is at most

S)

p(i _ p) 2 (,- 2 ) < In(logn)exp{-2logn+5(log•n)/n}

_ (log n)/n.

What is the probability that for some r, 3 <_ r _
7.1 The Connectedness of Random Graphs [n/2j r=3

165

/n/2]

rn r

2

prl(1

~P)r(n-r) <

[n/2 r=3

~r rg

_52p

log n ±rlog2

-r{logn+c+o(l)} + nlogn+c+o(1)} [n/2]


r 5 /2 exp{-2r(log n)/5} •< n/

r=3

This shows that a.e. Gp consists of a giant component and isolated vertices, so by Theorem 3.5, P(Gp is connected) = P{6(Gp) > 1} + o(1) = e-e - + o(1).

11

Theorem 7.3 shows that Mo(n) = (n/2)logn is a sharp threshold function for the connectedness of a graph: if M(n) = '{logn + wO(n)}, then a.e. GM is connected if wo(n) -* o and almost no GM is connected if co(n) --+ -oc. Kovalenko (1975) generalized Theorem 7.3 to the model W{n, (pij)}. Set qij = 1 - pij (so that qi 1 - pii = 0 for every i), n

Qik =

max

qj qij2 ... qij-,K

and 2

=

5 Qio.

Note that 2 is the expected number of isolated vertices. Kovalenko proved that if max Qio I <_i
-+ 0,

2

---

o

(7.6

and

Qik - (e'- 1)--+0

, k2 k=1

(7.6)

i=1

then a.e. graph in ý§{n, (pjj)} consists of a giant component and isolated vertices, and the number of isolated vertices tends to P;,o in distribution. Although in general relation (7.6) is not easily checked, it is satisfied if the probabilities pij are close to (log n)/n (see Ex. 3). The analogue of Theorem 7.3 for random r-partite graphs (see Ex. 5) was proved by Palfisti (1963, 1968). Considerably more-detailed investigations of the connectivity of random r-partite graphs were carried out by Kelmans (1967b) and Rucifiski (1981). Since connectedness is a monotone property, P(GM is connected) < P(GM+± is connected) for every M = M(n). It is fairly surprising that,

166

Connectivity and Matchings

as proved by Wright (1972) (see also Wright, 1973b, 1974c, 1975a), the probability of connectedness of an unlabelled graph can be less for more edges. We shall return to this in Chapter 9.

7.2 The k-Connectedness of Random Graphs The vertex-connectivity of a graph G is denoted by k(G) and the edgeconnectivity by .(G). Recall from Chapter 2 that, given a graph process G (G,)N, the hitting time 4(Q) = T(Q; G) of a monotone increasing property Q of graphs is defined as

z(Q) = z(Q; G) = mint : Gt has Q}. Thus z(Q; G) is the first time when the graph acquires property Q. In particular, z{K(G) > k} is the first time when the graph becomes kconnected and z{36(G) > kJ is the first time the minimum degree becomes k. Trivially z{((G) Ž> k;G} <_ z{K(G) _>k;G} for every graph process G. It is fascinating, and at the first sight rather surprising, that, in fact, equality holds for a.e. graph process. This was proved by Bollobfis and Thomason (1985) for every function k = k(n), 1 < k < n- 1, but here we prove it only in the case when k is constant. Theorem 7.4 Given k E N, a.e. G is such that 4(6(G) _ k) = z(k(G) > k). The proof is based on a lemma concerning separators. Given a graph G, a set S c_ V(G) is called a separator if V(G)\S is the disjoint union of two non-empty sets, say W1 and W2, such that G has no Wt - W 2 edge. Our notation will always be chosen to satisfy W,I IW2 ý.If S1 = s, we call S an s-separator. A separator is trivial if W1 = 1. Note that every non-complete graph of minimal degree s has a trivial s-separator but it need not have a non-trivial s-separator. Lemma 7.5 Let k > 2, w(n) --* oo, (o(n) •_ log log log n and p - {log n +

(k-1) log log n-w(n)}/n. Then almost no G, contains a non-trivial (k -I)separator. Proof Denote by Ar the event that Go contains a (k - 1)-separator with IW, = r. Set r0 = max{k, 10}, rl = [n 5/ 9J and r 2 = [(n-k + 1)/2J. The

7.2 The k-Connectedness of Random Graphs

167

lemma claims that Pp

A)

o(1),

or, equivalently, P

Ar

+P +)

UAr

U

+P

Ar

=0(1)

(7.7)

In estimating the first probability above we shall make use of two facts. (a) A.e. Gp has minimum degree k - 1. This follows by Theorem 3.5. (b) Given 1 e N, a.e. Gp is such that no two vertices of degree at most 1 are at distance at most 1. Indeed, the expected number of paths of length k > 1 ending in vertices of degrees i and j is at most nk+1pIpk(pn)i-l(pn)--(1 -_ p) 2n-i-j-2

=

0(1).

Now (a) and (b), with 1 = k + r0 , imply that p(Urr° 2 Ar) = o(l). To see this note that if 6(Gp) _Žk - 1 and Gp contains a (k - 1)-separator with IWI = r > 2, then Gp contains two vertices of degree less than k + r at a distance less than k + r. Let us turn to the estimate of the second term in (7.7). If 6(Gp) > k - 1 and S is a (k - 1)-separator with ro < IWI = r _
~)/ 1;i r( 7

(k-

)rkI)

Consequently, by (a), P (

U

n-r k-I1

n rr

o(1) +

Ar rr

)-

4 r(k-1)/1 (1

3

_ p)r(n-r-k+l)

=r rl

<

o(1)

nnr+k-3r(k-1)/14-r+2r 2 /n

+

.

o(1).

r=ro

Finally, a sufficiently good estimate of the third term in (7.7) follows from the fact that there are no W 1 - W2 edges:

P

(

2

A r) r~rrr

<,

--

r

n ) r

n( --r (I k--1

_- p)r(n--r-k+l)

168

Connectivity and Matchings eny nk(

r2

P)r(n/2 -k/2)

en r r=rt r2

r~rl r2

<

Z

n2 kn-r/ 20

=

o(l).

LU

r=rI

Equipped with Lemma 7.5, Theorem 7.4 is easily proved. Proof of Theorem 7.4 (i) Suppose k = 1. In this case we do not even need Lemma 7.5. Set wo(n) = log log log n, M = [1{logn - wo(n)}j and M2 = [n{logn + ow(n)}j. We know that a.e. G is such that z{r(G) Ž 1} > M 1, GM has at most log n isolated vertices and all other vertices of GM, belong to the giant component. Stop such a process at time M1 and denote by Z = {zi, .. ,z,} the set of isolated vertices of GM,. Thus 1 < s < log n. Now run the process for M 2 - MI more steps. The probability that one of the next M 2 - M, edges joins two vertices of Z is at most 2s2 (AM2 - M) ) n2o()

(1)'

This shows that, with probability 1 - o(1), every edge incident with some zi and drawn not later than time M 2, joins zi to the giant component of GM,. On the other hand, a.e. GM2 is connected. Hence z{TK(G) > 1}

=

{16(G) _>1} with probability tending to 1. (ii) Suppose k >Ž 2. In this case our theorem is an immediate consequence of Lemma 7.5. Indeed, by Lemma 7.5 and Theorem 2.2 there is a function M(n) such that M(n) < (n/2){logn + (k - 1)loglogn log log log n} and almost no GM contains a non-trivial (k - 1)-separator. Hence a.e. graph process G (G,)N is such that for t > M(n) the graph Gt does not contain a (k - 1)-separator. To complete the proof all we have to note is that z{5(G) > k} > M(n) for a.e. graph process and a graph with minimum degree k is k-connected unless it contains a non-trivial (k - 1)-separator. Let us note two more immediate consequences of Lemma 7.5 and Theorem 3.5. The first is due to Ivchenko (1973b) and the second to Erd6s and R~nyi (1961b). -

Theorem 7.6 If p(n) <_(log n + k log log n)/n for some fixed k, then P{K(Gp) = ý(Gp) = J(Gp)} -+ 1.

7.2 The k-Connectedness of Random Graphs

169

In fact, Theorem 7.6 holds without any restriction on p, as shown by Bollobfis and Thomason (1985). Theorem 7.7 If k E N and x e RiI are fixed and M(n) = (n/2){log n + kloglogn+x+o(1)} E N, then

P{x(GA•) = k}

-> 1 - ee

and

P{Ic(GM)= k + 1}

* e-e -/'!.

[

The moral of these results is that the connectivity of a random graph is not higher than what it is only because single vertices can be separated by their neighbours from the rest of the graph. The next result, from Bollobfs (1981a), shows that this is especially true for random graphs with constant probability of an edge. Theorem 7.8 Let p be fixed, 0 < p < 1. Then a.e. Gp contains t = [n1/7] distinct vertices x1 , x 2 . . xt such that if Go = G, and Gi = Gi- 1 - {xi}, 1, 2,..., t, then K(G) < i(G

1) -

t < K(G2 ) - 2t <... < K(Gt) - t2 .

Furthermore,for each i, 0 < i < t, we have K(Gj) = 6 (Gi) and Gi has larger minimum degree and larger vertex connectivity than any other subgraph of 7 order n - i. In 1995, Cooper and Frieze introduced the kth nearest neighbour graph and studied its connectivity properties. To define this random graph Ok, we start with the complete graph K, with vertex set, and pick a random order el ... , eN of the edges of Kn. For each vertex i, let Ei be the set of the first k edges incident with i, and define Ok = ([n], U Ej). The random graph Ok has a beautiful description in terms of random weights' on the edges. Assign i.i.d.r.vs from an atomless distribution to the edges of K. Then, with probability 1, no two edges have the same weight. To get the edge set of Ok, for each vertex take the k lightest edges. There is a 'dynamic' description of Ok as well, based on a random graph process (Gt)' or, what is the some, a random permutation el, e2 , eN of the edges. Define a sequence of graphs Ho ( H 1 cz ... c HN with vertex set [n] as follows. Set E(Ho) = 0. Having defined H,, if at least one of the two endvertices of e,+± has degree less than k then set

170

Connectivity and Matchings

E(Ht+i) = E(Ht) U {e,+l}, otherwise set E,+1 = E,. The process stops when every degree of Ht is at least k. The random graph Ok is rather close to Gk-OUt; for example, kn/2 <_ e(Ok) •! kn- (k+ ). As pointed out by Cooper and Frieze (1995), a.e. Ok has about 3n/4 edges, a.e. 02 has about lln/8 edges and, in general, for k = o(log n), a.e. Ok has

kn

-

2n - 1) l:_i<j
i-2 2d(i,)

-i+ j j - _2n

+Ok3 + 0(

~n Io gn)

edges, where d(i, j) is the Kronecker delta. This is easy to deduce from the martingale inequalities Theorems 1.19 and 1.20, provided one can calculate the expectation Concerning the connectivity of Ok, Cooper and Frieze (1995) proved that a.e. 01 is disconnected and, for every fixed k > 3, a.e. Ok is kconnected. It is rather surprising that 02 behaves in a somewhat peculiar way: for wo(n) -+ oc, a.e. 02 has a giant componant with at least n - w)(n) vertices. Furthermore, lim P(0 2 is connected) = r, n--*co

where r is a constant satisfying 0.99081 _ 0, let Gr(Xn) be the graph with vertex set X, in which two vertices are joined by an edge if their distance is at most r. As r increases, Gr(X,) has larger and larger minimum degree and connectivity. Given X,, let Pn,k = pk(X,) be the minimal value of r at which Gr(Xn) has minimal degree at least k, and define Un,k = ak(Xn) similarly for k-connectedness. Formally, the hitting radii S,,k and U,,k are given by S,,k = Sk (X,) = min{r > 0 :

3 (Gr(Xn)) > k}

:

k (Gr(Xn)) Ž k}.

and Unk = 0-k (X,) = min{r > 0

Extending his result from 1997, Penrose (1999) proved that if X" is a random n-subset of the d-dimensional unit cube Id then, with probability tending to 1, for the geometric random graph Gr(X,) these two hitting

7.3 Matchings in Bipartite Graphs

171

radii coincide: lim

P

(0n,k = Sn,k) = 1.

n--o0

note that this result is the exact analogue of Theorem 7.6. However, we should not be misled by the easy proof of that result: the proof of this extension to geometric random graphs is much harder.

7.3 Matchings in Bipartite Graphs In this section we shall consider the probability spaces j{K(n,n);p} and ýV{K(n, n); M} consisting of bipartite graphs with vertex sets V1 and V2 with VIl = IV 21 = n (see §1 of Chapter 2). In the first model we select edges independently and with probability p, in the second we take all bipartite graphs with vertex classes V1 and V2 which have M = M(n) edges. We write Bp and BM for random elements of these models. Furthermore, we define a random bipartite graph process B = (Bt)" 2 in the natural way. Our aim is to find a threshold function for the existence of a complete matching. In other words, we are interested in the hitting time r(match; B) of a matching in our bipartite graph. Once again, it is trivial that c(match;B) > z(6(B) > 1; B) for every bipartite graph process B since the minimum degree is at least 1 if the graph has a complete matching. Our main aim is to prove that we have equality for a.e. G. This result, from Bollob~is and Thomason (1985), is a slight extension of a theorem of Erd6s and R~nyi (1964, 1968). It is worth remarking that z{K(B) > 1; B} = z{T(B) > 1; B} also holds for a.e.

B (see Ex. 4). In proving that two hitting times agree a.e., it is useful to introduce some other models of random graphs. Although in the first application, which is the main result of this section, we shall work with bipartite random graphs, it is more natural to define the more commonly applicable models, namely §(n,p; > k) and ý%(n, M; > k), as in Bollobfis (1984a). Both models consist of graphs with vertex set V, whose edges are coloured blue and green. To define a random element of ýN(n,p; > k), first take an element H of W,, i.e. select edges independently and with probability p. Let xt, x2 ... , x, be the vertices of degree less than k in H. For each i, 1 < i •s s, add to H a random edge incident with xi. Colour the edges of H blue and the edges added to H green. (So there are at most s _ k). The model W(n, M; _>k) is constructed analogously.

172

Connectivity and Matchings

A little less formally: a random element of ,f(n,p; > k) is obtained from a random element of S(n,p), coloured blue, by adding one directed green edge ry for each vertex x of degree less than k and then forgetting the direction of the edges (and replacing multiple edges by simple edges). Lemma 7.9 Let k e N be fixed and let Q be a monotone increasing property of graphs such that every graph having Q has minimum degree at least k. Let p = {logn + (k - 1)loglogn

-

o(n)}/n

where wo(n) --* oc and w(n) < log log log n. If a.e. graph in 91(n, p; >_k) has Q, then T(Q;G) = r{T(G) > k;G} for a.e. graph process G. Proof Since a.e. graph in W(n,p;> k) has Q, by Theorem 2.2 there is a function M1(n) G N such that n

logn + (k -

l)loglogn - 32 w(n)

_ Mi(n)

n {logn+(k-1)loglogn-

Io(n)

and a.e. graph in N(n, M 1 ; > k) has Q. Let M2 (n) = [(n/2){logn+(k-1)loglogn+wo(n)fl and let 4. be the set of graph processes G = (Gt)' for which 6(GMl) = k - 1, 6(GM2 ) > k and no edge added at times M1 + 1, M1 + 2. M2 joins vertices which had degree less than k in GM,. It is easily checked (cf. §1 of Chapter 3) that a.e. GM, has minimum degree k - 1 and at most log n vertices of degree k - 1 which form an independent set. If an edge is added to such a graph, then the probability that the edge joins two vertices of degree k - 1 is at most 4(log n) 2/n 2. Hence, with probability at least I - 10(log n) 2 wo(n)/n, no edge added at times M 1 + 1, M 1 + 2, ... , M2 will join vertices of degree less than k in GMý. Furthermore, by Theorem 3.5, a.e. GM 2 satisfies 6(GM•) = k. Consequently P(4.) = 1 - 6n, where -+ 0. Let S. be the collection of graphs from W(n, M1; > k) in which the blue graph has minimum degree k - 1 and no (blue or green) edge joins two vertices of blue degree k - 1. Once again, rather trivially, P(IN.) = I -v,, where En - 0. Let us define a map 0

S .*, as follows. Given G = (Gt)N

e

•,

7.3 Matchings in Bipartite Graphs

173

let O(G) be the coloured graph whose blue subgraph is GMH and whose green edges are the first edges added after time M, and not later than M2 which increased the degree of a vertex to k. Clearly iO(S.) = S. and (1 - En)P{ -'(A)l = (1 - &n)P(A) for all A c S..

(7.8)

By assumption the set o= {G

S(n, MI; - p) :G has Q}

is such that P(10) = 1 - q/, where

qn --+

0.

Hence P(OO n SO) > 1 - gn - fln. and by (7.8) the set J' = P ' )ý

n S.) satisfies 1 -- 6n(1 -E 1 -en

-tJ

--

This completes the proof since by our construction if G k(G) has Q], then

-Y [and so

z{6(G) > k; G} = r(Q; G).

D

It is very important (though rather trivial) that the probability measure in S(n,p; >_k) is fairly close to the measure in S(n,p). Lemma 7.10 Let A and B be (disjoint) subsets of V(2). Then the probability that a graph G in S(n,p;>_ k) is such that Ac-E(G)

and BnE(G)= 0

is at most (p

A (1

-p)IBI.

Proof Note simply that the probability that a set A 1 green edges is at most {2/(n - k)} A.I

=

V(2) consists of

Now we are well prepared to prove the main result of the section. Theorem 7.11 A.e. random bipartite graph process B is such that r(match;B) = z{r(B) > 1}.

174

Connectivity and Matchings

Proof Set p = (logn - logloglogn)/n and give SI{K(n,n),p;> 1} the obvious meaning. By the analogue of Lemma 7.9 for bipartite graphs, it suffices to prove that a.e. graph G in §{K (n, n), p; > 1} has a matching. In our estimate of the probability that a graph GeS{K(n,n),p; > 1} has no matching we shall make use of the following immediate consequence of Hall's theorem (see, for example, Bollobas, 1978a, pp. 9 and 52; 1979a, p. 54). L Lemma 7.12 Let G be a bipartite graph with vertex classes V1 and V2, I /Y1 1 = IV21 = n. Suppose G has no isolated vertices and it does not have a complete matching. Then there is a set A ( Vi (i = 1,2) such that: (i) F(A) = {y : xy G E(G) for some x E A} has JAI- 1 elements, (ii) the subgraph of G spanned by A U F(A) is connected and (iii) 2 _<JAI <--(n + 1)/2. Proof As G does not have a complete matching, Hall's condition is violated by some set A ( V., i.e. F(A)J < AI,.

(7.9)

Choose a set A of smallest cardinality satisfying (7.9). Then (i) and (ii) hold, since otherwise A could be replaced by a proper subset of itself. If, say, A c V1, then B = V 2 - F(A) is such that F(B) (- V1 - A, and so B also satisfies inequality (7.9): IF(B)I <

IV - JAI < IV2 1-

F(A)I = IBI.

Hence JAI -
1F(A)l = n - (JAI1), so JAI _<(n + 1)/2. Let us continue the proof of Theorem 7.11. Denote by F, the event that there is a set A c- Vi(i = 1 or 2), JAI = a, satisfying (i), (ii) and (iii) of Lemma 7.12. All we have to show is that in !N{K(n,n),p;>_ 1} we have P

where nl

=

UFa

= o(1),

[(n + 1)/2J.

Let A1 V1,A 2 C- V2 and JAII = [A 2 I + 1 = a. What is the probability that the subgraph of G spanned by A1 U A 2 has at least 2a - 2 edges and no vertex of A, is joined to a vertex in V2 -A 2 ? By the analogue of Lemma 7.10, very crudely this is at most a(a - 1) 2a -2 ) (ng)

2a-2,+l)

(1-

)logn

7.3 Matchings in Bipartite Graphs

175

Since we have 2 (") choices for A with ýA= a and (afor F(A),

(n

ZP(F)<2Z

nn a

-Pa a=2

a=2

a(a -1) 2a -2)

1

a-a2

< 2z--,\a __ n,(en

(aY--)en

a(

(log n

-2n

(22)

a-1 (ea_2a-2(logn

2

1

)

more choices

a- 2

(1 -p)a(n-a+l)

2a-

(loTn

a=2

2a-2

-a n (loglogn og

)-a+az/n

n

nl

_ E(e

E]

log n)3anl+a+a 2/n = o(1).

a=2

In view of Ex. 6 of Chapter 3, we have the following immediate consequence of Theorem 7.11. Corollary 7.13 Let xeR, M = (n/2){logn + x + o(1)}eN and write a random element of !{K(n, n), M}. Then

BM

for

P(BM has a matching) -- * e2

We conclude this section with a brief account of a problem about matchings in randomly weighted complete bipartite graphs, the random assignment problem. The set-up is much like in the minimum weight spanning tree problem, discussed in §5 of Chapter 6. Let X, = (Xij) be an n by n matrix whose entries Xij, 1 • i, j < n, are non-negative i.i.d. r. vs. Let G(Xn) be the n by n bipartite graph with vertex classes V = {vl,...,vu} and W = {wl .... wn}, in which viwj has weight X1 . What is the minimum weight m(X,) of a complete matching in G(XJ)? In other words, what is the minimum weight of a one-to-one assignment V -* W? If the Xij are uniformly distributed in (0, 1), then we write Un= (Uij)

instead of X,

=

(Xij) and call the problem the (uniform)

random assignment problem. Donath (1969) was the first to consider this problem and to notice experimentally that the expectation E(m(U,)) seems to tend to some constant C* between 1 and 2. Walkup (1979) proved that the expectation was indeed bounded, in fact, limrnE(m(U,)) < 3. He deduced this from his result that a.e. n by n bipartite 2-out graph has a complete matching. Walkup's method was further developed by Walkup (1980), Frenk, van Houweninge and Rinnooy Kan (1987), Karp (1987), Dyer, Frieze and McDiarmid (1986) and Karp, Rinnooy Kan and Vohra (1994). In particular, Karp (1987) lowered Walkup's upper bound to 2.

176

Connectivity and Matchings

Concerning a lower bound, trivially, n

E(m(Xn)) > ZE(min Xij) = nE(min Xij). j~1

J

J

In particular, for the uniform random assignment problem, E(m(Un))

n

+ I-

The first non-trivial lower bound was given by Lazarus (1993), who proved that lim,_E(m(U,)) > 1 + 1/e. Goemans and Kodialam (1993) improved the lower bound to 1.44 by extending the methods of Lazarus, and Olin (1992) to 1.51. It is clear that if we are interested in the limit of the expectation E(m(Xn)) then only the distribution of Xij near 0 matters. In particular, we may replace Uij by A1j, where Aij has exponential distribution with mean 1, i.e., P(Aij < x) = 1 - ex for x > 0. Aldous (1992) observed that considering the exponential random assignment matrix A, = (A11) has many advantages over the uniform random assignment matrix U, = (Uij), and many of the proofs become much simpler. Among other results, Aldous (1992) proved that c* does exist: the expected cost of a minimum assignment (with the exponential distribution, say) does tend to a constant c* between 1.51 and 2. Let us see how the lower bound 1 + 1/e given by Lazarus (1993) can be proved if we use exponential r. vs A/j, 1 < i, j < n, each with mean 1. Set Ai = mini Aij and Bij = Aij -Ai. Then each Ai is exponentially distributed with mean 1/n. Furthermore, the entries of the matrix B, = (Bij) are again i.i.d. exponential r. vs with mean 1 except that a random element Bij(i) of each row is replaced by 0. For 1< j
E(m(A,))

n

E E(Ai) + E E(B3j) i~l

ji-1

since a matching has an element in each row and column. Hence 1 E(m(An)) > n 1 + E(2)/n = 1 + E(Z)/n, n where Z is the number of columns of B, containing no 0. Indeed, if the jth column of B, has a 0 then E(BRj) = 0, otherwise Bj. is the minimum of n independent exponential r. vs, each with mean 1, so E(Bj) = I/n.

7.3 Matchings in Bipartite Graphs

177

Write Sr for the expected number of r-tuples of columns of B, containing no 0, and set Pk = P(Z = k). Then

Sr=

(

) (n -r)n,

and by the inclusion-exclusion formula (1.19), n-i

n-i n-i

E(Z) = Zkpk = k=l

=-

/~~ r

n

YZk(-r k

r

\

)\

n

n n

k=1 r=k

n-I

(

E(_lr r=1

r Y

n~ )jr(1

I)kk(

r

=n (n-)n.

k=1

Therefore, E(m(An)) _ I + ( n

'

and so lim E(m(An)) > 1 + 1/e. n ---

cQ

Mesard and Parisi (1985, 1987) used the 'replica method' of statistical physics to show that c* = 7r2/6. Although the replica method is not mathematically rigorous (at least, not yet), its predictions are frequently 2 correct. Note that 7T /6

-

Fk_' k-

2

= C(2) is reminiscent of the value

3

C(3) = k- proved by Frieze (1985a) for the minimum spanning tree problem (see §5 of Chapter 6). After over a decade's stagnation, Coppersmith and Sorkin (1999) made progress with the upper bound: by analysing an appropriate assignment algorithm, they proved that c* < 1.94. Shortly after that, in a remarkable paper, Aldous (2001) proved that c* is indeed 7r2/6. However, this is not the end of the story. For from it. Parisi (1998) made the staggering conjecture that E(m(An))

k

= k=t

for every n. Here, as before, An = (Aij) is the exponential random assignment matrix: P(Aij • x) = I - ex for x 2! 0. Parisi's conjecture seems to be far from a proof, but Coppersmith and Sorkin (1999) gave an ingenious generalization of it. Let A(n, m) = (Aij) be on n by m matrix whose entries are i.i.d. exponential r. vs, each with mean 1. Let mk(A(m, n)) be the minimal sum of k entries, no two of which are in the name row

178

Connectivity and Matchings

or column. Coppersmith and Sorkin (1999) conjectured that 1 E(mk(A(m, n))) X-" (m - i)(n - j) i,jŽOi+j
for every k, I < k < min{m,n}. Note that for m = n = k we have mk(A(m, n)) = m(A,,) and the Coppersmith-Sorkin conjecture implies the conjecture of Parisi. Coppersmith and Sorkin (1999) proved some partial results that indicate that their conjecture is indeed true, and Alm and Sorkin (2001) proved the conjecture for k < 4, k = m = 5 and k = m = n = 6. For a wealth of information about the random assignment problem, see Coppersmith and Sorkin (1999) and Alm and Sorkin (2001).

7.4 Matchings in Random Graphs At what stage t will a.e. random graph Gt of even order have a complete matching? Since the minimum degree of a graph with a complete matching is at least 1, this t has to be at least (n/2){log n + wo(n)}, where w(n) -* c. The somewhat surprising fact that this trivial necessary condition is also sufficient was proved by Erd6s and R~nyi (1966). In this section we shall deduce this theorem from a sharper result: a.e. graph process on an even number of vertices is such that the edge raising the minimum degree to 1 also ensures that the graph has a complete matching. More importantly, as proved by Bollob~is and Thomason (1985), when a r.g. has only slightly more than (n/4) log n edges, then with probability tending to 1 it has a matching covering all but at most one of the vertices of degree at least 1. A further strengthening of this result was proved by Bollobits and Frieze (1985). We start with the last result mentioned above. As usual, we work with the model S(n,p). What can we say about p (p > 1/n, say), if a.e. Gp has a matching covering all but at most one of the vertices? If Gp is such a graph, then it does not contain six vertices: x1 ,x 2 ,x 3,x 4,y, and Y2, such that each xi has degree 1, x, and x2 are joined to yl, and X3 and X4 are joined to Y2. Indeed, if we have six such vertices, then every matching misses at least one of xl and x 2 and at least one of X3 and X4. Now the expected number of configurations formed by X1 , x 2 and yl is

(

)

(n

-

2)p 2 (1 _-p)2(n-3).

It is easily seen that if 1/n <<_p < (1/2n)(logn + 2loglogn + c) for

7.4 Matchings in Random Graphs

179

some constant c, then the expectation is bounded away from 0 and, furthermore, with probability at least E(c) > 0, the graph contains such a configuration of six vertices (Ex. 7). Hence p has to be at least (1/2n){logn + 2loglogn + wo(n)}, where wo(n) -* cc. The main result of the section shows that this trivial necessary condition is sufficient. Theorem 7.14 Let p = {logn + 2loglogn + wo(n)}/2n, where wo(n) -4 Oo. Then a.e. GP has a matching covering every vertex of degree at least 1, with the exception of at most one vertex. Proof Let p0 = (log n + log log n)/n. Since a.e. GN 0 is connected, we may and shall assume that p < po. The proof consists of a series of lemmas. The first three have no obvious connection with matchings. Let A, be the event that (i) Gp consists of a giant component and at most n 1/2 / log n isolated vertices, (ii) it has at most n"/ 2 vertices of degree 1, and (iii) no two vertices of degree 1 have a common neighbour. D Lemma 7.15 P(A1) =1 + o(1). Proof By the remark after Theorem 7.2, the trees of order 2 vanish after time (n/4){log n + log log n + wo(n)}. Furthermore, it is immediate (cf. the proof of Theorem 7.3. 1) that the expected number of vertices of degree 0 is o(n 1/ 2/ log n) and the expected number of vertices of degree 1 is o(nl/ 2). Finally, p was chosen exactly to ensure that the expected number of pairs of vertices of degree 1 having a common neighbour tends to 0:

(

) (n - 2)p 2 (1 p)2(n-3) "

(

2logn +o(n)

exp{-logn-2loglogn-

o(n)} =o(1)

Let A2 be the event that Gp does not have r0 = (4 log log n/ log n)n independent vertices.

Lemma 7.16 P(A2) = 1 + o(1). Proof The expected number of ro-tuples of independent vertices is n ) (1 ro

p)ro(ro-l)/ 2 <

l-

r e ro

pro(ro-I)/

2

.

(7.10)

Connectivity and Matchings

180 Since

1

e logn

2en

2

loglog n

ro 0

the right-hand side of (9) is o(1).

Let A 3 be the event that the complement of Gp does not contain a K(ro, to), i.e. a complete bipartite graph with r0 vertices in each class. Lemma 7.17 P(A3) = 1 + o(1).

(r,)

Proof The two vertex classes of a K(ro, ro) can be selected in 2 (n-ro) < (;) different ways. Hence, as in the proof of Lemma 7.16, the expected number of K (ro, ro) graphs in the complement of Gp is at most 2 en)

0(1 _p)r2 <

r0

\ en rod

2

e0

<-2r-r }r

= o(1).

D

<

The combinatorial basis of the proof is the extension of Tutte's factor theorem (1947) due to Berge (1958) (see Bollobis, 1978a, pp. 55-57). Call a component of a graph odd if it has an odd number of vertices, and denote by q(H) the number of odd components of a graph H. Then no matching covers all but at most one vertex of a graph H if there is a set R ( V(H) such that q(--R)

R +2.

(7.11)

For r = 0, 1.... denote by Br the event that there is a minimal set R ( V such that R1 = r, and Gp - R has at least r + 2 components which are joined to R and each of which is either an isolated vertex or has at least three vertices. Furthermore, let B(r, s) be the event that Br holds and the union of the r + 1 smallest components in the definition above, which we shall denote by S, has s elements. Note that if s < r then Br = 0. We see from (7.11) that if A1 holds for Gp and Gp does not satisfy the conclusions of Theorem 7.14, then Br holds for some r, 0 < r < n/2. Hence, by Lemma 7.15, Theorem 7.14 is proved if we show that P

P([n/2J

Br) = o(1).

(7.12)

r=h If Br holds, then Gp contains a set of r + 2 independent vertices:

7.4 Matchings in Random Graphs selecting one vertex each from r + 2 components of Gp such a set. Hence

181 -

R we obtain

ln/21

U Br c A2, r=ro

so by Lemma 7.16 relation (7.12) follows if we show o(1).

P ( r,=Br)

(7.13)

Let A 4 be the event that one cannot find a set R c V such that 1 _< IRI < r0 , Gp - R has at least two components and the union of all but a largest of the components of Gp - R has at least r0 vertices. Lemma 7.18 P(A 4 )

=1 +

o(1).

Proof We claim that A3 (- A4, so the assertion follows from Lemma 7.17. To see A 3 c A 4 , suppose Gp - R has components of orders a, _• a2 :< ... _< a, and ZI- ai > r0. Let j be the smallest index with Zi., ai > ro. Then Zai < r0 + aj <_ ro + (n-- ro)/2 = (n + ro)/2 i=1

so

Saia_

n - r0 - (n + ro)/2 _>r

i=j+l

Since no edge joins one of the first j components to the others, the complement of Gp contains a (•ai,

K (

1

ai

la

a)

- K(ro, ro).

Clearly

Ur-I Br,

U

J

J r=l s=r+l

B(r, s)

and

{JiB~r~si} (nA4

=0

Ii

182

Connectivity and Matchings

so by Lemma 7.18 relation (7.13) and our theorem follow if we show that

U Our next aim is to replace Set so = (log n) 3 n1 / 2 .

B(rs)

(7.14)

=o(1).

U..,+ 1 B(r, s) by a considerably smaller union.

Lemma 7.19 Relation (7.14) holds if

0=1 =S,+iB(r,s))

=

(7.1t5)

0(1).

Proof We have (,)(n-r) choices for R and S. The minimality of R implies that the R - S edges and the edges joining vertices of S form a connected subgraph of Gp. Hence there are at least 2r edges from R to S. Finally, no vertex in S is joined to a vertex not in R U S. Therefore if 1•r < s • r0 , then

n

s)) P{5B (r,

)(n

enr

,ns

r where no

=

-

r)

rs )

P2 ,(1 _ ps(n-r-s)

'es~2r

e-P•° = B(r, s),

n - 2ro.

By the definition of p and no e-Pno _ (n)(log n)3 n-1/12, where E(n)-

O.Hence if so :< s _< ro and r < s, then 2 B(r, s + 1)/!(r, s) •_ (en/s)e e-pn°

This implies that with so(r)

=

.

<

2

max(r, so) we have

ro

Z

B(r, s) <_2B{r, so(r)}.

ss(r)

Therefore if (7.15) holds, then so does (7.14), provided r0

I:B(r,r)= o(1). r=s0

7.4 Matchings in Random Graphs

183

The last relation is immediate since ro

ro

3B(r, r)

,

I: Ze

r=s0

p

2r

(P2n)

p

re-Pr

r

0o< r

(

o

• E{elon

r•s 0

)

2

n~l

11

.

E

r=so

For h > 1 let Ch be the event that V contains pairwise disjoint sets T,H and S such that 0•< ITI = t < so-h, IHI = h,3 •__ ISI = s <so, S spans a component of Gp - T U H, each vertex of T U H is joined by an edge to some vertex in S and for each x G T there is a vertex y which is isolated in Gp - T U H and is adjacent to x. Lemma 7.20 P(Ch) = o(1) for every h E N. Proof The probability that a set S of s vertices spans a connected subgraph is at most

S p

s-1 Hence the expected number of triples (T, H, S) with the parameters above is at most

(n( ) n ) )2

n

i

t~p t~l _p)

t

lo

(esp)slI(Sp)llh(l


Ps, n)2 t+s+hSt+hn

{n-2(lgn)3}t+s(lg

s+t

et! I st+hnl t/2s/2 (log n)5t+4s+h

C(s, t),

where c, is an absolute constant. Now if s < t, then 2 C(s, t + 1)/C(s, t) < cin-1/ (log n)

5

and if t < s, then C(s + 1, t)/C(s, t) •_ ce2hn-1/2 (log n) 4 . Consequently I

C(s,

C(3, t)

t) <

s=3 t=s=3

t=O so

1: (ecI) 2sShnl-S(log n) 9s+h + O{n- 1/2 (lognr)h+ 12 }

=

0(1),

st3

which completes the proof of the lemma.

L

184

Connectivity and Matchings

We have arrived at the last step in the proof Theorem 7.14. Let h0 > 6 be fixed. Given t > 0, u > h0 and w > 3u, denote by C(t, u, w) the event that with r = t+u and s= t+w the event Br holds, R= TUU,S = T' U WIT = IT = t,U = u and W1 = w, the set T' consists of vertices isolated in Gp - T U U, Gp contains a complete matching from T to T', each vertex in T is adjacent to at least one vertex in W, W consists of u + 1 components of Gp - T U U and each of those components sends at least h0 edges to U. Recalling that if A1 holds then no vertex has two neighbours of degree 1, SO SO

h0

s0

U U {B(r, s)lnAýlUCU r=1 s=r+1

h=O

so

SO

U U C(t,,u,w). t=O u=ho w=3u+3

Consequently, by Lemmas 7.15 and 7.19 our theorem is proved if we show that So

SO

SO

SS S

P{C(t,u,w)}

(7.16)

=o(1).

t=O u=ho w=3u+3

To prove (7.16) we make use of the fact that Gp[W] has at least w-u-1 edges and U and W are joined by at least uho edges. We obtain the following crude upper bound on P{C(t,u, w)}: n;2~f(-p Wt )P(n) (n)(w(w- 1),2) Pw-u-I WPh(1-)" t!

tnPo'

Wt

t

W

t~2 !ll(p +u+W(pn) tw t {n- i/

2

=

UW ho

(log n)3}t+ w °

,,-U u+w

ho pw//ho)u V plU 1 U- 1(pw op'-

(t, u, w),

where cl is a positive absolute constant. Clearly

C(t, 1 + 1, w)/C(t, u, w) <_ cn (pw)ho < n-1, pw

so for w < so < nl/Z(log n)

3

SO

5 C(t, u, w)

< 20(t, ho, w)

u=ho

Ilc•+W(log n)5t+4w+4hn-t/2-w/2+2ho+l-h/2/wt


where c2 depends only on ho.

h

185

7.4 Matchings in Random Graphs Therefore So

So

t

So

t

E E E C(t, ho, w) •_ E E(logn)12tn-t/2-w/2-2 t=O u=ho w=ho

=

o(1)

t=O w=ho

and SO

S SO

3 •3

Z0 C(t, ho, w)<

t=0 u=ho w=t

SO

SO

5 Z

wtnt/2 -t/4

2 -

0(1).

t=O w=t

Relation (7.14) is a consequence of these inequalities so the proof of El Theorem 7.14 is complete. The proof of Theorem 7.14 was rather cumbersome because we had to consider graphs with only about (n/4) log n edges. As we know, about twice as many edges are needed to make it likely that a r.g. has no isolated vertices. In that range the proof would have been considerably simpler, as would a direct proof of the following immediate consequence of Theorem 7.14, which is an analogue of Theorem 7.3. Corollary 7.21 (i) Let x G IR,n = 2m and M = (n/2){logn + x + o(1)}.

Then lim P(GM has a 1-factor) = lim P{6(GM) > 1} = 1 -e m---L

m-'c

(ii) If n is even, ow(n) --* oo and M = (n/2){log n + wo(n)} then a.e. GM

has a 1-factor.

El

Not much effort is needed to prove the hitting time version of Corollary 7.21: one checks that the estimates in the proof of Theorem 7.14 sail through for W6(n,p; > 1) with p = (logn - logloglogn)/n and then applies Lemma 7.9. Theorem 7.22 A.e. graph process G on an even number of vertices is such that z(G has a 1-factor;G) = r{j(G) > 1;G}

=

{IK(G) > 1;G}.

If we do not insist on a complete matching and are satisfied with a set of (n/2) + o(n) independent edges, then many fewer than (n/2) log n edges will do. Theorem 7.23 Let M = noo(n), where wo(n) -* oo. Then a.e. (n/2) + o(n) independent edges.

GM

contains

186

Connectivity and Matchings

Proof This is an immediate consequence of Berge's extension of Tutte's factor theorem and the fact that for any fixed t > 0 almost no GM contains [enj independent vertices (Ex. 8). El Theorem 22 tells us that the obstruction to a 1-factor is the existence of vertices of degree 0. What happens if we take only those elements of §(n, M) which do have minimal degree at least 1? BollobAs and Frieze (1985) proved that, perhaps not too surprisingly, the critical value of M is about n4lognl+ nloglogn. Theorem 7.24 Let M = (n/4)(log n + 2n log log n + c,), n = 2m,m e jV'. Then 0, lim P{GM has a

Ic- {Ifactor(GM) im

1}=

e -e-'/

if' c, --* -- 0 8

sufficiently slowly, , if cn, c,

1. 1,

if c,

-0*

0.

The condition that c, -ý -- o sufficiently slowly seems curiously out of place but, in fact, some condition is clearly needed for if, rather crudely, M = m = n/2, then P{GM has a 1-factor IJ(GM) > 1} = 1 since 6(GM) = 1 implies that

GM consists of m independent edges. In spite of the strong resemblance between Theorems 7.14 and 7.24, the results are very different since Theorem 7.24 claims that a.e. element of a set of very small probability has a matching, while Theorem 7.14 is about a.e. element of the entire space. There is a function m(c) such that if c > 0 is a constant and M - cn/2, then a.e. GM is such that the maximal number of independent edges in GM is {re(c) + o(1)}n. Unfortunately, at present there is no closed form expression for m(c). Karp and Sipser (1981) gave simple algorithms which with probability tending to 1 find {m(c) + o(1)}n independent edges in a r.g. GM. Earlier, Angluin and Valiant (1979) gave a fast algorithm which for a sufficiently large constant c finds complete matchings in .(2m, clog n/n) with probability at least 1 - O(n- 2). For large values of c we have rather tight bounds on re(c). Since a.e. Gc/, has about e-'n isolated vertices, m(c) < (1 - e-c)/2. Frieze (1985d) proved that this upper bound gives the correct order of m(c): e-c _•1 - 2m(c) < {1 + 8(c)}e-c, where lim...

er(c) = 0.

7.4 Matchings in Random Graphs

187

A k-factor is a natural generalization of a complete matching: a kfactor of a graph G is a k-regular subgraph H of G whose vertex set is the entire vertex set of G. Most of the results concerning complete matchings can be extended to k-factors. The next result, due to Shamir and Upfal (1981a), generalizes Corollary 21(ii). Theorem 7.25 If k Ž_ 1, kn is even, w(n) --+ o and M = (n/2){logn + E, -o(n)}, then a.e. GM has a k-factor. (k - 1)loglogn +

The hitting time version of Theorem 7.24 is an immediate consequence of a theorem of Bollob~is and Frieze (1985): a.e. graph process is such that for even n the very edge increasing the minimum degree to k ensures that the graph contains k independent 1-factors. It is interesting to note that Shamir and Upfal proved Theorem 7.25 by the method of alternating chains rather than by making use of Tutte's f-factor theorem. Shamir and Upfal (198 lb) also proved a variant of Theorem 7.24 for graphs with at least n(logn)wo(n) edges, where

wo(n)

-- oo.

Aronson, Frieze and Pittel (1998) greatly sharpened the Karp-Sipser results on matching algorithms we mentioned above. For example, they proved that if c > e then in GI/ the Karp-Sipser algorithm almost merely finds a matching whose site is within n1/ 5 +o(l) of the optimum. Frieze, Radcliffe and Suen (1995) ran the following very simple matching algorithm on random cubic graphs. At each stage, choose a vertex u of minimal degree from the current graph, pick a neighbour v of u at random, add uv to the matching under construction, and delete u and v from the current graph. Frieze, Radcliffe and Suen proved the rather surprising result that if we write 2, for the number of vertices not matched when this algorithm is run on a random cubic graph G3-reg, then the expectation of 2, satisfies cin1/ 5 < E(A,)

<_C2n 1/5 logn

for some positive constants cl and c2. Dyer and Frieze (1991), Dyer, Frieze and Pittel (1993), Aronson, Dyer, Frieze and Suen (1994, 1995) and others analysed the performance of randomized matching algorithms ran on fixed graphs. For example, Dyer and Frieze (1991) studied the randomized greedy matching algorithm in which the next edge is always chosen at random from those remaining. They showed that there are graphs G for which this algorithm usually constructs a matching with hardly more than gl(G)/2 edges, where

188

Connectivity and Matchings

oc.(G) is the maximal number of edges in a matching in G. (Trivially, every matching that cannot be enlarged has at least oci(G)/2 edges.) On the other hand, Aronson, Dyer, Frieze and Suen (1995) proved the rather surprising result that if we change the algorithm slightly by picking first a non-isolated vertex of the current graph and then its neighbour, then the expected site is at least (1 + e)cli(G) for some fixed e >0. Rather different aspects of random matchings were studied by Kahn and Kim (1998) and Alon, Rbdl and Rucifiski (1998). Thus, Kahn and Kim proved that if in a fixed r-regular graph G we consider a matching M selected at random from the set of all matchings (not only maximal matchings) then for large r the probability that a given vertex v is not on an edge of M is about r-1 /2 Following Alon, Rbdl and Rucifiski, call a graph of order 2n super (oc,e)-regular if it is bipartite with classes V1 and V2, where IV I= IV 21 = n, such that the degree of every vertex is at least (c-e)n and at most (o(+e)n, and if U - V1 and W c V2 with Uý >- Fn and WJ >- en then e(u, w)

e(V1,

V2 )

Alon, R6dl and Rucifiski (1998) proved that if 0 < 2e < Y < 1 and n > n(e) then every super (c.,e)-regular graph of order 2n has at least (c - 2e)'n! and at most (a + 2e)nn! complete matchings. Note that for r = r(n) > 2en almost every r-regular biportite graph satisfies these conditions. The rest of this section is somewhat of a digression from the main line of this book; we shall give a brief account of some results about 1-factors and large matchings (sets of vertex disjoint hyperedges) in uniform hypergraphs. Using Chebyshev's inequality, de la Vega (1982) proved the analogue of Theorem 7.23 for hypergraphs: if r > 2 is fixed and M = now(n), where w(n) -* oc, then a.e. r-uniform hypergraph with n labelled vertices and M edges has n/r + o(n) vertex disjoint hyperedges (see Ex. 9). It is considerably more difficult to investigate 1-factors of hypergraphs. Schmidt and Shamir (1983) proved that if r > 3 is fixed, n is a multiple of r, wo(n) --* oo and M = n3 / 2 co(n), then a.e. r-uniform hypergraph with n labelled vertices and M edges has a 1-factor, i.e., has n/r vertex disjoint edges. By applying the analysis of variance technique first used by Robinson and Wormald (1992) in their study of Hamilton cycles (see Chapter 8),

7.5 Reliable Networks

189

Frieze and Janson (1995) greatly improved this result: for every 8 > 0, a.e. 3-uniform hypergraph on [n] with M(n) = [n 4/3+'j edges (3-tuples) has a 1-factor. It seems that this difficult result is still far from best possible: it is likely that the threshold for a 1-factor is again the same as the threshold for minimal degree at least 1. Thus the following extension of Corollary 7.21 is likely to be true. Let X c R and M = M(n) = n(log r + x + o(1)). Then if n -* oo through multiples of r, the probability that a random r-uniform hypergraph on [n] with M edges has a 1-factor tends to 1 - e- e. In fact, with probability tending to 1, the hitting time of a 1-factor may well be exactly the hitting time of minimal degree at least 1. Cooper, Frieze, Molloy and Reed (1996) used a similar technique to prove surprisingly precise results about 1-factors in random regular hypergraphs. Fix r > 2 and d > 2, and set Pd = 1 +(logd)/ ((d - 1)log d-_) Let

be a random d-regular r-uniform hypergraph with vertex set [n], where r divides dn. If r < Pd then a.e. G(r) has a 1-factor, and if r > Pd then almost no G(r) has a 1-factor. The proof requires rather difficult moment calculations. Finally, we turn to a beautiful and deep theorem of Alon, Kim and Spencer (1997) about near-perfect matchings in fixed graphs. There are constants c3 , C4, .... such that the following holds. Let H be an r-uniform d-regular hypergraph on n vertices such that no two edges have more than one vertex in common. Then, for r = 3, H contains a matching covering all but c3nd- 1/2 (logd)1/ 2 vertices and, for r > 3, there is a Gd(r) Vd-reg

matching missing fewer than Cknd-1/(r-l) vertices. In a Steiner triple

system on n points, every point is in d (-1) 2triples, so this theorem implies that every Steiner triple system on n points contains at least n/3 - c 3n'/ 2 (log n) 31 2 pairwise disjoint triples.

7.5 Reliable Networks Graphs are unsophisticated models of communication networks. Adopting a rather simplistic view, one can say that a communication network should carry messages between pairs of centres as reliably as possible. Each communication line has a certain probability of failing, and one is interested in the probability that in spite of the failure of some lines it is still possible to send a message from any centre to any other centre. In the simplest case it is assumed that each communication line transmits information both ways and that the lines fail independently and with

190

Connectivity and Matchings

a common probability. Thus one is led to the model ,f(H;p)and the question: what is the probability that a random element of W(H;p) is connected? Recall that a random element of Sg(H ; p) is obtained by selecting edges of H independently and with probability p. In other words, we have a communication network H in which the lines fail independently and with probability q = 1 - p. Let us write fH(P) for the probability that a random element of S&(H;p) is connected. If H is a graph with m edges then, trivially, fA(P) = Em fjpJ(1 -- p)m-j, j=n-1

where fj = fj(H) is the number of sets of edges of cardinality j which contain a spanning tree of H. This form of fH(p) was first investigated by Moore and Shannon (1956). By now there is a formidable literature on the problem of finding exact or asymptotic information about this polynomial: Birnbaum, Esary and Saunders (1961), Kelmans (1967a, 1970, 1971, 1976a), Lomonosov and Polesskii (1971, 1972a,b), Polesskii (1971a, b), Boesch and Felzer (1972), Frank, Kahn and Kleinrock (1972), Van Slyke and Frank (1972), Lomonosov (1974), Margulis (1974), Bixby (1975), Even and Tarjan (1975), Rosenthal (1975), Valiant (1979), Ball (1980), Buzacott (1980), Ball and Provan (1981a,b), Bauer et al. (1981), Boesch, Harary and Kabell (1981) and Provan and Ball (1981). The papers in the list above, which is far from complete, tackle a variety of aspects of the reliability problem: they study the computational difficulty of finding some or all the coefficients of the polynomial fH(p), analyse the cut sets and spanning trees of H, choose a graph with as many spanning trees as possible (provided the order and size are fixed) and discuss how suitable subcollections of spanning trees and cut sets can give reasonable bounds on fH(P). Instead of saying a few superficial words about each of the papers above, we shall concentrate on a beautiful theorem of Margulis (1974), which belongs to the pure theory of random graphs. This result concerns a threshold phenomenon in the model ýý(H; p). Let H, be a connected graph of order n > 1. What can we say about the function ftj(p) as n --+ oo? Since connectedness is a monotone increasing property, by the remark after Theorem 2.1 the function fH,(P) is strictly increasing, fH,(O) = 0 and fH,(1) = 1. Will fH,(P) increase from being almost 0 to being almost 1 in a short interval? In other words, given

7.5 Reliable Networks

191

0< e < , does f -1(1 - t) -f-(e) tend to 0 as n --* oc? If it does, then the sequence (H,) behaves like (Ku), the sequence of complete graphs, namely there is a threshold function po(n) E (0, 1) such that if p is slightly smaller than po, then fH,(P) is close to 0 and if p is slightly larger than p0, then fH,(P) is close to 1. Not every sequence has a threshold function for connectedness. As a simple example, let H, consist of two complete graphs K n and K"2 = K"-n joined by an edge, where n, = [n/2]. Then fH,(p) -+ p uniformly on [0,1], since the probability of H, being connected depends mostly on the probability of the edge joining KnI to Kn2. Clearly, if we join Kn1 and Kn2 by a fixed number of edges, then the obtained sequence (H,,) still fails to have a threshold function for connectedness. On the other hand, if we join Knl and Kn2 by l(n) edges and 1(n) -+ oc, then (H,) does have a threshold function. Note that in the latter case the edge-connectivity 4(Hn) of Hn tends to oo as n --+ oo. What Margulis proved is exactly that 2(H,) -+ oo ensures that the sequence (H,) has a threshold function for connectedness. Theorem 7.26 There is a function k : (0, 1) x (0, 1) -* R such that if 1,(H) >_k(6,e), then fH-1(1 - o) - fH-1 (8) < 6. Theorem 7.26 is a consequence of a more general theorem of Margulis (1974), which concerns set systems. Let X be a finite set of cardinality m and let sl be a collection of subsets of X : V : Y(X). Endow Y(X) with the Hamming distance: d(A,B) = AABI and define the boundary OV of d as 0,/= {A c Define a function h,

: Y(X)

: BB 0 ýV,d(A,B) -

Z+ by

hd(A) = fl{B ý V : d(A,B) = 1}1, hU(A 0,if

1}.

if A c , A 0 sl,

and for 0= sl ý £6(X) set k(.d) = min{h (A) : A c OV1z}. Call sV a monotone set system over X if A c B c X and A (EV imply B c _.V. Given 0 < p < 1 and a set X, select the elements of X independently and with probability p. Denote by fw(p) the probability that the set formed by the selected elements belongs to sV. If sl is monotone and

192

Connectivity and Matchings

0 =# V = J(X), then (cf. the remark after Theorem 2.1) f.01 (p) is a strictly increasing function, fi, 1 (O) - 0 and f

=,(1) 1.

Theorem 7.27 There is a .function k : (0, 1) x (0, 1) a monotone system and k(sl) >_ k(6, e), then

-*

R such that if .4 is

0) -- f''l (8) < 3

f0/ 0--

Let us see first that Theorem 7.27 implies Theorem 7.26. Let k(6, r) be the function given in Theorem 7.26 and let H = (V, E) be a graph with 2(H) >_ k(, 8). Set X = E and &I = {A Then ,

A c E and H - A is disconnected}.

is a monotone system and 1 - fH(P) =f(1

p). p-

Hence all we have to show is that k(V') _>(H).

(7.17)

Let AEcas. Then H - A is disconnected and there is an edge ceF - A such that (H - A) + c%is connected. Consequently, H - A consists of two components, say with vertex sets V1 and V 2 = V - V1 , and h,,(A) is the number of V, - V2 edges of H. Thus h.(A) _ 2(H), proving (7.17). We shall not present the entire proof of Theorem 7.27, but we shall show how the specific conditions on V are used. Denote by lp the measure on 9(X) defined by our independent selection of the elements: for BEgP(X) put pp(B) = p JB(1

-

p)m-IBI,

where m = IXI,

and for M (_ ŽA(X) put PP(s)

p(B).

=

Bc-4

Note that f,I(p) = up(V). Furthermore, set

=th Byp() ea dfntnoh By the definition of h.•,, we have qWp(V4) >_

hoI(A)dup >! h(_4)1tp(OWd).

(7.18)

7.5 Reliable Networks

193

Lemma 7.28 Let sl be a monotone family. Then for 0 < p < 1 df•j(p)

_ Wp(sl)

dp

p

Proof For 0 _
Y(X) :BI= k} and

k=

Vnk.

Furthermore, given W, - - Y(X), write [W, -q] for the number of pairs (C,D) such that C = D,C e 16 and D c -9. Clearly

Z

h4,,(A)

=

[Vk,&Vk-1 --

=

kj

k-1] = [-lk,.Xk-1] -

k1-(m-k+01)J

[Qluk,Slk-1]

k-li.

Therefore with the convention s1_1 = 0 we have WpP(,)=f

h (Aid

-P=Z

k=O

h(dkp

p

k=O m

AC

k

k-)1}

m

pk+(1

=ZklWkIpk(1 _p)m-Wk k=O

=P

rfk(1-m-k h

k=O

, k

p)m-k{kl4kl -(m n-lk + O

pk(l

-

f

_ p)m-k-1

k=O

Ik lpk(1-_p)m,-

=pk=0 kýO

I

k

p(m -k)

-

\-m

k

km(pn (p

1-

df.,(p) dp

In addition to inequality (7.18) and Lemma 7.28, the proof of Theorem 7.27 hinges on the following theorem about general set systems. (In fact, one only needs the result for monotone set systems.) For a proof, which is far from easy, the reader is referred to the original paper of Margulis (1974). Theorem 7.29 There is a continuous strictly positive function g : (0, 1) x (0,1) --* (0,1) such that if Q1 ( (X),0