Periodica Mathematics Hungariva Vol. 10 (1), (1979), pp. 47--53

L I M I T T H E O R E M S F O R C O M P L E T E S U B G R A P H S O F

R A N D O M G R A P H S

by

K. SCH-~RGER (Hoidelberg)

In the sequel all graphs in consideration are finite, undirected and have neither loops nor multiple edges. We study the random graphs I'p ~ I'p(n) being defined as follows. The set of vertices of I'p(n) is { 1 . . . . , n}. Each possible edge is chosen with probability p -~ p(n) C [0, 1] and different edges occur independently. ~or many problems there is no essential difference between this type of random graphs (already mentioned in [2]; compare also [5], [3]) and that one in which a number N = N(n) of edges is chosen at random such that

all ({2N)) possible choices are equiprobable if p(n) is chosen in such a way tha~

p ( n ) { 2 ) = N ( n ) ( s e e [ 2 ] ; for another type of random graphs compare [11]).

I f F is a family of graphs, denote by Pp{ I'p(n) 6 F} the probability that Pp(n) belongs to F. We study the behaviour of I'p(n) for large n. In this context we ~re interested in complete subgraphs (subcliques) of random graphs (a graph G is called complete if G has every pair of its vertices adjacent). A k- clique is a complete graph with k vertices. Denote by zk(n) the number of k-subliques of Yp(n). We would like to mention that k-subcliques of random graphs in which p does not depend on n have been studied in [6], [7] and [9].

I f G is a subgraph of G', we write G c G'. By C we denote suitably(l) chosen positive constants depending only on parameters but not on n.

LE~cIA 1. Let G 1 . . . . . Gr be pairwise diHerent k-cliques (for fixed k ~ 2) being not all pairwise vertex-disjoint. Let the graph G 1 U �9 �9 �9 [J Gr have Vr vertices and er edges. Then we have

A M S (MOS) subject olassl]ioations (1970). Primary 60F05, 05C30. Key words and phrases. Central limi~ theorem, enumeration of graphs.

4 8 SGH~RGER:C0~PLETE SUBGRAPHS OF RAND0~ GRAPHS

1

(1) V r g r , k > 2 , r > 2 .

~ (:}

(This upper bound is best possible. To see this consider the ease t h a t only two of the graphs a : , . . . , Gr are not vertex-disjoint and have exact ly one ver tex in common.)

The inequal i ty (1) improves a result in [10].

P~oo~" of L e m m a 1 (by induct ion on r). :First let r = 2 and assume t h a t G 2 -- O 1 has h (1 ~ h < k -- 1) vertices. Then clearly

v~2 = k + h k -- c

for some number c ---- c(h). This leads to

(~ h) hk 1 c(h) - ~ c(~ - 1) = : - , : < h < k - - 1.

( k - - 1) (k + h) + h(k -- h) 2

Hence (1) is proved for r -~ 2. Now assume (1) to be t rue for some r ~ 2. L e t h be the number of vertices of Gr+l -- (G: U �9 �9 �9 U Gr). We m a y assume 1 ~ h ~ k -- 1. Clearly

(2) V r + : = V r + h ' e r + : ~ G ~ { : ) - - ( k - - h l 2 , ' l ~ h ~ k - - 1 .

I f the graphs G: . . . . . Gr are not pairwise vertex-disjoint , we can apply the induct ion hypothesis and get by (2)

Vr+~ < V r + h ~ v r + h

~ .. +

1 1 k---- k

r r G 1

I f the graphs G 1 , . . . , Or are pairwise vertex-disjoint ,

1 k

Vr+ : ~ r k + h _~. r + 1 l ~ h ~ k - - 1

'r,-.

S C H O R G E R : C O ~ P L E T E $ U B G R A P H S OF R A N D O N G R A P H S 49

If ~ is a random variable, denote by E(~) and V(~) expectation and vari- ance, respectively, of ~ (it will be clear from the context which probabilities are involved). Let Kk(e) denote the family of all k-cliques over { 1 , . . . , r~}.

Pu t for O 6 Kk(~)

(3)

Introduce

(4)

L ( a ) = [ 1, G c Fp(n)

! O, otherwise.

T,(~, ~) = X E (L(O~) . . . % ( a A )

the summation extended ever all combinations (of order r) of graphs G~ E Kk(n ) without repetitions such that G1 . . . . . ~r are not pMrwise vertex- disjoint. Now we can prove

(5)

LElmgA 2. For k ~ 2, r ~ 2 and n ~ rk we have

k-I (1 I \~ rk-w~-.. .-wr 1 Tr(k ' ~) ~ r ] k----~. " ~ X

l ~ w a + . �9 +wt~rk -k- I

X J_~I-](kr-l l I ( j k - w 2 - ' ' ' - w j ) j=~ - - wi+1) . w)+l

P~oor. Consider different graphs Gj ~ Kk(n), 1 ~ ~" < r, which are not pairwise vertex-disjoint. Let w i denote the number of vertices of (G~ U �9 �9 �9 U U G j - O f'I Gj, 2 <~ r ~ r. T h e n G 1 U . . . U Gi has j k - - we - - . . . - - w i ver t i " ces. Let vr and er be defined as in Lemma 1. According to (1)

er ~__ 1 1

k---- k---- T r

k - - l ( l + r k l - ~ l ) ( r k - - w ~ 2 . . . . . Wr)"

I~ence

1

Tr(k, n) << r! O~w~,~.~'w,~k 1 ~W2 +.. �9 +wr~rR-k- 1

+o. + .... "

k - - w r

-- Wr--ll X

and this immediately yields (5).

4~ Perlodir l~sth. I (IO)

50 SCH~KGEK:COMPLETE SUBG~APHS OF ~ANDON GRAPHS

Now we can prove

T~EOR~M 1. Assume that (lc ~ 2)

(6)

exists and put

(7)

Then

(8)

2

lira p(n) n k-1 -= ~ E (0, c~)

lira Pp {~k(n) = s} ---- - - e-~-, ' 8 = 0, 1, . . . ,

i.e., the number z~(n.) o/ l~-subeliques of _Pp(n) asymptotically has a Poisson distribution with mean #.

PaOOF. Denote b y ~ (n ) the number of all ]c-subeliques of Tp(n) which are ver tex-dis joint to oJi other subgraphs G ~ Kk(n) of IPp(n). B y (5) and (6)

(9) P~{~(~) # ~(n)} = oO)

(the symbols o(1) and 0(1) will be used only for the limit n ~ ~ ) . Define for a graph G E Kk(n)

(10) I~(a) = t 1,

[ 0,

Pug

and

G (2_ Fp(n), and G is ver tex-dis joint to alJ o ther graphs G' C Kk(n), G" ~ l~(r~) otherwise.

Sr(]c, n) = X E(In(G~) . . . In(Gr)), r ~ 1

s;(t:, ~) = X ~ ( I ; ( G 1 ) . . . I;(Gr)), r > 1

4- 1 the summat ions ex~endea over Mi combinat ions (of order r wi thout repeti- tions) of graphs Gi E Kk(n), 1 ~ i ~ r, being pMrwise vertex-disjoint . We h a v e

(11) 0 :~ St(k, n) .... S;(]~, n) _~_ (r ~- 1)Tr+l(k, n), r ~ 1

and hence by (5) and (6)

(127 S'r(k, n) = St(k, n) -{- o(1), r ~ l .

SCH~RGER: COMPLETE SUBGP~APHS OF RANDOM GRAPHS 51

N o w

{;)( ' ( '); r

=r-~. ( b l ) ' ( n - - r b ) ! p "~r-! . l~. 0 J r ! ' r 2 1 "

= ~ . Therefore by (12) lim S~(k, n) r ! ' r ~ 1. An e~sy application of Bon-

ferroni's inequalities (see, e .g . , [1], p. 25) yields lira Pp {~k(n) = s} = ~{e -~,

s = 0, 1 . . . . This, together with (9), leads to (8).

COROLLARY 1. For k ~ 2 we have

I 1, lira p(n) ~k-I _= 13

(13) lira Pp {uk(n) = 0} = n - - n ~ 2

0, lira p(n) u~-f = ~ .

2

This fact showing that A(n) ----- ~k-I (k ~ 2) is a "threshold function" for the property of the random graph Fp(n) to contain a k-clique, could have been guessed from a result in [2]. Threshold functions for the random graphs considered in [11] have the property that they depend only o n t h e number of edges but not on the number of vertices of the graphs in question. This is in sharp contrast to the random graphs considered in [2].

Theorem 1 says that Uk(n) asymptotically has a Poisson distribution with mean #(/c, ~) if (6) holds. Now i~ is well known (see, e.g., [4], p. 194) that if random variables X~, 2 ~ 0, have a Poisson distribution with mean 2, X~ -- 2

V~ asymptotically (i.e., for ~ -~ ~ ) has. a normal distribution N(O, 1)

(mean O, variance 1). Hence we are led to consider the case that

(14)

Now

if we define

(i) (15) a = a(n, k, p) = ~ . p ,

2

lira p(n) rok- l = ~ . n - ~

rlr - - 1) ar(n ' k, p), ~ ~ rk,

k ~ 2 , n > r k .

4*

52 8CH~GE~:COMPLETE SUBGRAPHS OF RANDOM GRAPHS

Hence for each fixed r ~ 1, k ~ 2

(16) n > rk.

Assume additionally tha t for a f ixed k > 2

2 (17) lim p(n) n k-I = O, for all ~ > 0

(i.e., the convergence to infinity in (14) is not "too quick"). Then by (5)

" o(n T,(k, n) = 0 = o =

and we get, using (11) and (16),

((1)) 1 0 a~(n,k, , . ( i s ) s ; (k , n) = 7. , 1 + p) + o ( n - ~ - ~ ) r >_ 1

Now we can prove

T~EORE~ 2. Let k ~ 2 be [ixed and assume that (14) and (17) hold. Then, i f a(n, k, p) is de/ined as in (15),

X

(19) lira ?p [ ?a~ ,~ - ,p~ ~ x = 1 / ~ e U ~

2 d u , - - ~ < x < ~ .

1

Paoos . Since Tr(]C, n) ---~ o(n- ~), r >~ 2 we have Pp {nk(n) r ~k(n)} = o(I). Hence it suffices to prove (19) for ~ (n ) instead of ~k(n). Since the normal distr ibution N(0, 1) is uniquely determined by its moments it suffices to prove

1 J (20) n-,-~lim E [ ? ~ 0 ~ - " = ~ ~r e 2 d g , ?" = 1, 2 , . . .

(compare [8], p. 185). I f S(r, ]) denotes the Stirhng number of t h e second type (giving the number of unordered partit ions of a set of r elements into ] nonvoid sets),

r

(21) E@~(n)) = . ~ j ! S ( r , j ) ~ ( k , n ) , r 2 1. j=l

Therefore by (18) and (17)

E(~;(n)) = o ( ~ - ~ ) + s(r , j ) a~(n, ~, p), r > 1,

S~H~RGER:COMPLETE S~BGRAPHS OF EANDO~ GRAPHS 5 3

which gives

E ~ k ( n ) - - a r = o ( 1 ) q - a -r]~ ( - - 1 ) r a ~ q - ~ ( - - 1 ) ~ a ~ S ( r v - i , j ) a , i=0 j=l

(22) r : 1 , 2 , . . . .

S ince ( compare [2], p. 33)

x ] ( 2 3 ) ~ ~(r,j)X j -~- ~ ~. e-x,r

j=l j=0 we ~rr ive ~t

r - - 1, 2 1 . . . ,

(24) 1 2 , . . . .

Because o u r a s s u m p t i o n s i m p l y l i m a ( n , k , p ) = ~ , the r i gh t side in (24) n * - b ~

conve rges to t he r - th m o m e n t o f N(0 , 1) (see [2], p. 33), a n d (20) is p r o v e d .

I~EFEt~ENCES

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[2] P. ERD6S and A, R#.NYI, On the evolution of random graphs, .~agyar Tud. Alcad. Mat. Kutat6 Int. K6zl. 5 (1960), 17--61. M R 23 ~= A2338

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[10] K. SCH~2~ER, ~ber die Entwicklung zu]~lliger p-2oartiter Graphen, Universit/~t Heidelberg, 1974. (Unpublished thesis)

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(P~eceived August 9, 1976)

DEUTSCHES KRE BSFORSCHUNGSZENTRUM INSTITUT FOR DOKUMENTATION, INFORS:[ATION UND SS:ATISTIK IM NEUENHEiMER FELD 280. D--69 HEIDELBE]~G FEDERAL REPUBLIC OF GERMANY