Random Graphs: 1st, 1983: Seminar Proceedings (Annals of Discrete Mathematics 28)

RANDOM GRAPHS '83

NORTH-HOLLAND MATHEMATICS STUDIES

118

Annals of Discrete Mathematics (28)

General Editor: Peter L. HAMMER Rutgers University, NCJW Briiirswick, N J , U.S.A.

Advisory Editors C. BERGE, Universite' de Paris, France M A . HARRISON, University of' California, Berlieley, CA, U.S.A. V. KLEE, University of Washington, Seattle, W A , U.S.A. J.-H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. G.-C. ROTA, Massachusetts Institute of Technology, Cambridge, M A , U.S.A.

NORTH-HOLLAND- AMSl ERDAM * NEW YORK * OXFORD

RANDOM GRAPHS ’83 Based on lectures presented at the 1st Poznan Seminar on Random Graphs, August 23-25, 1983, organised and sponsored by the Institute of Mathematics, Adam Micltiewicz University, Poznan, Poland.

Edited by

Michal KAROQSKI

and Andrzei

RUCII’;SKI

A d m i Mickiewicz University

Poznari, Poland

1985

NOR.rH~HOLLAND- AMSlERDAM . NEW YORK . OXFORD

0 Elsevier Science Publishers B. V.,

1985

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 878211

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Library of Congress Catalogiu~in Publicnlion Data Main entry under title: Random Graphs ’83. (Annals of Discrete Mathcnratics; 28) (North-Holland mathenlatics stitdies; 118) “Sponsored by the Institute of Mathematics, Adam Mickiewicz University, Pornad, Poland” A selected collection of papcrs based on lectures presented a t the 1st Poznari Seminar on Random Graphs held a t Adam Mickiewicz University, Poznari, Poland, in August 1983. 1. Random Graphs--Probabilistic Methods in Combinatorics--Addresses, essays, lectures. I. Karonaki, M. (Michnl), 1946-. 11. Rucinski, A. (Andrzej), 1955-. IIJ. lnstitute of Mathematics, Adam Mickiewicz University, Pomah. Poland. IV. Series. V. Series: North-Holland mathematics studies; 118. ISBN 0-444-87821I

PRINTED IN POLAND

PREFACE Random graphs, as a separate area of research, emerged in the form of the series of the fundamental works of Erdos and RCnyi in the early sixties. However, the history of this new branch can be traced back to the applications of probability methods to combinatorics originated by Szekeres, TurBn, Szele and Erdos i n the forties. In recent years random graphs have attracted, and still continue to attract, a considerable interest all over the world. This is documented by ever growing number of papers in this area and published in the variety of combinatorial, probabilistic as well as general mathematical journals. It was for want of an international forum on random graphs that we decided to suppleinent our Pozna6 Seminar with biennial mcetiiigs, thus providing a platform for an exchange of ideas between iiiatheniaticians working in this field. The opportunity of starting meetings arobe while the International Congress of Mathematicians (ICM) was held i n Poland in 19S3. The first seminar, to which a representative group of researchers was invited, took place i n August 23-25, 1983 in Poznali at Adam Mickiewicz University. Forty mathematicians from twelve countries attended the sessions. Profcssors Paul Erdos and Jocl Spencer delivered invited plenary lectures, and Professor Wiadyslaw Orlicz, the Honorary President of the ICM Warsaw meeting, was our special guest. A highly stimulating, informal atmosphere during the meeting strongly contributed to a fruitful exchange or opinioiisaswell as to the initiation of many professional contacts. The prcsent volume covers a wide scope of random graphs topics such as structure, colouring, algorithms, mappings, trees, network flows, and percolation. Papers in this collection also illustrate the application of probability methods to Rainsey’s problems, the application of graph theory methods to probability, and relations between games on graphs and random graphs. All the papers presented, some by authors who for various reasons were not able to take part at the seminar, were subject to a refereeing process. We are very grateful to all referees for their outstanding contributions. We also wish to acknowledge the substantial help provided by Adam Mickiewicz University, the sponsor of the meeting, and Professor Andrzej Alexiewicz, Director of the Institute of Mathematics. Our sincere thanks are also due to the General Editor and the Publishers of this series for their encouraging support. Poznali, September 1984 Michal KAROSSSKI and Andrzej RUCIfiSKI V

CONTENTS

PREFACE

V

...

LIST OF PARTICIPANTS

Vlll

P. ERDOS,Welcoming address

1

J. BECK,Random graphs and positional games on the complete graph S. BERG,A note on randoill mappings: Convolutions and partitions B. BOLLOB~~S and A. M. FRIEZE, On niatchings and hamiltonian cycles in random graphs B. BOLLOBAS and A. THOMASON, Kandoni graphs of small order M. DONDAJEWSKI, P. KIRSCHENHOFER and J. SZYMANSKI, Vertex-degrees in strata of a random recursive tree W. GAUL,Reliability-estimation in stochastic graphs with timc-associatcd arc-set reliability performance processes G. GRIMMETT, Electrical networks with random resistances 1. JAWORSKI, A random bipartite inapping G. 0. H. KATONA,Probabilistic inequalities from extrenial graph rcsults (A survey) A. D. KORSHUNOV, A new version of the solution of a problem of Erdos and Renyi on harniltonian cycles in undirected graphs K. KUULASMAA, Locally dependent random graphs and their use in the study of epidemic models L. MUTAFCIEV, A random sanipling procedure froin a finite population and some applications J. NESET~IL and V. RODL,TJiree remarks on dimensions of graphs Z. PALKA, Bipartite complete induced subgraphs of a random graph A. RUCINSKI, Subgraphs of random graphs: A general approach I. H. SMIT,Matchmaking between two collections J. SPENCER, Four roads to the Ramsey function J. L. SPOUGE,Random graph problems in polymer chemistry

7

vi

15

23 47 99 107

125 137 159 171 181

189 199 209 22 1 23 I

243 25 1

Contents

W-C. S. SUEN,Flows through complete graphs 1. TOMESCU, On the number of trees having k edges in common with a caterpillar of moderate degrees W. F. de la VEGA,Random graphs almost optimally colorable in polynomial time K. WEBER,Subcube coverings of random graphs i n the n-cube P. WHITTLE, Random graphs and polymerisation processes J. C. WIERMAN, Critical percolation probabilities

vii

263

305 311 319 337 349

LIST OF PARTICIPANTS J. Beck, Budapest, Hu11g~11.y S. Berg, Luntl, Sweden H. Rielak, Lublin, Poland M. Dondajewski, Poznati, Polrind P. Erdos, Budapest, Hungury Z. Furedi, Budapest, Himgary G. Grimnett, Bristol, Englmd J. Gruszka, Pomari, Polund S. Janson, Uppsala, Swcden J. Jaworski, Poznaii, Poland F. Juhasz, Budapest, Hiingary R. Kalinowski, Crucow, Poland M. KaroAski, Poznati, Poland G. 0. H. Katona, Budapest, Hzingury J. Knoska, Poznali, Poland K. Kuulasmaa, Oulu, Finland Z. Loiic, Warsaw, Poland D. W. Matula, Dallas, U.S.A. L. Mutafciev, Sojia, Bulgaria J. NeSetFil, Pragtie, C.S.S.R. Z. Palka, Poznati, Poland J. Raburski, Poznari, Poland A. Ruciriski, Poznali, Poland P. Sablik, Poznati, Polana' S. Samulski, Poznah, Poland M. Skowroriska, Toriiri, Poland Z. Skupieri, Cracow, Poland I. H. Smit, Anistcrdain, Holland E. Soczewiriska, Lublin, Poland J. Spencer, Stony Brook, U.S.A. J. L. Spouge, Oxford, Englanci W-C. S. Suen, Bristol, England L. Szaiiikolowicz, wroc/aw, Polcind J. Szyniariski, Poznak, Poland A. TruszczyAska, Warsaw, Poland M. Truszczyriski, Warsaw, Polanrl W. F. de la Vega, Paris, France B. Voigt, Bielefeld, West Germany K. Weber, Rostock, G.D.R. J. C. Wiernian, Baltimore, U.S.A. viii

This Page Intentionally Left Blank

Annals of Discrete Mathematics 28 (1985) 1-5 0 Els-vier Science Publishers B. V. (North-Holland)

WELCOMING ADDRESS PAUL ERDOS Matliernarical Itwritrite of the Ilungariaiz Academy of Sciences, H-1053 Biidfipesi, Iliiti,;vzry

It g;ves me great pleasure to g've this welcoming address to the first Poznari meeting on random graphs. Perhaps the aud'ence will forgive a very old Jllm to g.vc a few historical reminiscences liow I came to apply probability methods i n combinatorlal analysis. I niLi5t do this while my mind and memory are st 11 more or le55 intact. 1 do not intend to g ve a history of the probability method but will restrict myself almost entirely to my own contributions. I will start with liamsey's theorem. Denote by r ( u , u ) the smallest integer for which every graph of r ( u , u ) vertices either contains a complete p p h of u vertices or an independcnt set of u vertices. The well known proof ol Szekeres gives

( :y ' )

and in particular r(n , n ) q

r ( u , v)
1. Note that some restriction must be placed on the growth rate of 1c.l when c,+ - co as

Our second result is a generalization of one stated by Komlos and Szemertdi [13]. Tostatethisweneed todefine tliefollowing:agruphprocess611=(C,, G I , ..., GI,, ...) isaMarkovprocessinwhich C,, is a graph with vertices V,,=(l, 2, ..., H > and edges El,, where (E,,I=m. C, is obtained from by choosing e E V ~ 2 ) - E n l - uniformly I at random and putting E,,,=E,l,-l u { e } . Note that GI, above is distributed exactly as GI,,,. For a graph property Il (usually monotone) and graph process 6, let T ( T D)=min(m , :G,,,Eff).

In particular let I7,='The minimum degree of G is at least k' and has Lk/2] disjoint hamiltonian cycles plus a disjoint matching if k is odd.'

I?,='C

Our second result is

On ma f chings

and hamiltonian cycles

25,

Theorem 1.2. If k is a fixed positiue integer then lim P T ( T ( ~ 11,)=7(r, , I?,,>=]. ,I+",

K o ~ i i l 6and ~ SzemerCdi stated this result for k = 2 . Note that Theorem 1.2 is most clearly stated as: i f we randomly add edges one by one then when the graph constructed has minimum degree k then it a.s. has lk/2] disjoint IiamilLonian cycles plus a disjoint matching if k is odd. For other results on inatchings and haniiltonian cycles i n random graphs see Bollobis [2], Bollobh, Fenner and Frieze [4], Fenner and Frieze [7], [S], Frieze [lo], [ I I], [ 121, Richmond, Robinson and Wormald [ 141, Richmond and Wormald [15], Robinson arid Wormald [16], Shaniir [17], and Shamir and Upfal [lx], [19].

Notation For a graph G we let V ( C ) denote its set of vertices and E(G) denote its aet of edges. For U E V(C), dG(u) is the degree of o, and for SG Y(G), N , ( S ) = ( u # S: there exists w E S such that (0, w ) E E(G)). For non-negative x, V1,(G)=(vE V ( G ) :d G ( u ) > x } . For SE V(G),G[S]=(S, Es) where E S = ( e e E(G):e c _ S ) . Let D,=D,(G)'be the set o f vertices of degree 1 i n G and let y/(G)=G[V2(G) -NG(DJI. For e~ E ( C ) we let C - P = ( V ( G ) ,E ( G ) - [L')) and for e $ E(G) we let G + e =( V(G),E ( G ) u { e ) ) . For O < p < I , G,,>,denotes R random graph with vertices {1,2, ..., / I ) i n which each of the (:) possible edges is chosen with probability p and not chosen with probability 1 - p .

2. Throughout this section m=/z log n/4+t7 log log n/2+c,, / I where for the moment we assunie [c,,l+- m. The proof of Theorem 1.1. I > obtained by a sequence of lemmas.

Lemma 2.1. Let G = G,,,,, LARGE= VlognllOO(G)arid SMALL= V(G)-LARGE. Consider the followitiy condilirion.9: No cycle of length 3 contains 2 small vertices;

(2.la)

B. Bollobds, A. M. Frieze

26

No path of length 2 contains 3 small vertices; SG V(C), 4< IS1 < 7, IS n SMALL 3 implies G [ S ] is not connected; (SMALL]~ n - ~ ~ ; 9 # SELARGE, IS1

(2.1 b) (2.lc) (2.1 d) (2. I e) (2.1f)

Then for n large Pr(G,,, fails to satisfy ( 2 . l ) ) ~ n - * ~ 5 .

(2.2)

Proof (Outline). To estimate the probabilities for (2.la), (2.lb), (2.lc), (2.lf) we simply compute the expected number of triangles containing 2 small vertices, etc. This is tedious but stra'ghtrorward. To deal with (2.ld), (2.le) we let p=(logn/2+loglognf2cl,)/n and consider the random graph Gn.p. As ]E(G,,,)) is a binomial random variable with parameters and p , it is easy to verify that

e)

Pr (IE(G,,

,)I

= m)>,

+ ( n log n)-'

for n large.

A1 so G , , p conditional on IE(G,,,)l=m is distributed exactly as C,,,

. (2.4)

Thus for any property I7 Pr(G,l,nlhas IZ)2n'1210gn)~ Chernoff bound. To obtain (2.18) from (2.20) we define 3(d)={G:

V ( G ) = ( l , 2 , ..., r r + t } ,

IV,(G)I=n and IE(G)I=m'),

where we assume throughout that Im'-ml 6 2 d i 2 1 0 g n .

We note (2.21)

(2.22) as both sides of (2.22) count the number of pairs (C, e), where G E 3 ( d - I), e 4 E(G) and G + e E 3(m'). Now (2.21) implies r n ' ~ 9 ( ~ n f ) ~ > u(G)2(m'-n,(m'))(3(mf)l,

(2.23)

G E ??(rn')

where nl(m') is the expected number of vertices of degree 1 i n a random graph chosen uniformly from 9W). , Next let

,>I

A d = Pr(l V1(Gn+t,

=n )=

ImQ11

On matchings and hamiltonian cycles

33

It follows from (2.22) and (2.23) that (2.24) where

I n order to apply (2.24) to "close the gap" between mi and rnl in (2.20) we must estimate ii,(rn'). Weshow first that ifcr(c)=(e1-2c/2)(1+o (I)) then, wherep=(logn/2+loglogn +2ci)/n, C i - w ,

(I

I,,

Pr D,(G, +,,

(2.25)

2 Pn') G ( P b (c))-afl+

The above probability is no more than the probability that there exists s=IPor(c)n'''] vertices, each of which i, adjacent to at most one of the other n --s vertices. This latter probability is

which implies (2.25). We next prove the very crude lower bound

To do this, we proceed as in the proof of (2.19), using G,,+t,nl'in place of G n + , , pand , define events A and B. Now Pr(A)>(t/n)'(l-o(l)) as before but we cannot use the FKG inequality to bound Pr ( B A ) which is Pr (6(Gn,,,l-)21). Instead, let now p=log n/2n and then

I

Pr(6(Gn,,)>1)GPr(6(G,,,,)B

t)+Pr(lE(G,,,)I>m').

We then use the FKG inequality as before to get a lower bound

The Chernoff bound gives

(2.27)

B. Bollobds, A . M. Frieze

34

for n large. Using these inequalities in (2.27) gives P r ( 6 ( G , , , m . ) ~ 1 ) ~ e - ” f / 4for n large. This is easily good enough to prove (2.26). Now (2.5), (2.25) and (2.26) together imply

(2.27) Putting p=max (2, ar ( c ) e) in (2.27) we easily obtain (2.28) Using (2.28) in (2.24) we see that for large

ti

(2.29) where 0 depends only on c. (2.20) and (2.29) together imply the lemma.

0

For the remainder of this section t is as in Lemma 2.6. Now let

Now Lemma 2.5 implies Pr(,u(Gi,’i) = Ln/21)= Pr(X1Y).

However, it follows from Lemma 2.4 (with n + t in place of n) and Lemma 2.6 that P r ( X n YnZ)/Pr(Y)a

<e-

<e-e2

r

+f - k ) ) 2 r n - k a - r

+ -k )

(Zm -ka)2 613 (n f

lognli3

for n large.

Thus Pr (there exist more than s=n/log log n vertices of degree exceeding a)

A similar argument deals with vertices of degree less than 2 (1 - 8 ) m/n.

Proof of (2.39b). To prove (2.39b) we need to be able to generate a random G E &'3(d) with probability

(note that this is not the same for all G E &'3(d)). We modify the method of Bollobsis [I]. Thus, let d E Q, be fixed and let W,

W,,

..., W,+,be disjoint sets with IW,]=difor i = l , 2 , ..., n+t.

n+l

Let W =

U W;.

i= 1

and let the members of W be denoted as points. A configuration F is a partition of Winto m pairs of points called the edges of F. Let c be the set of possible configurations and note that Icl=N(rn)=(2m!)/m!2"'. For P E W, let p(p)=i, for i= 1,2, ..., n+t and for F E C let p(F) be the rnultigraph ({1,2, ..., n+t), { ( p ( p ) , ~ ( 4 ):){ p , q } E F ) ) . Note that p(r)=AS(d). We turn into a probability space by giving each F E the same probability. This induces the required probability space on p(c). (Think ofgerierating MG,+,,, conditional on MGn+,,,E & Y ( ~ )by taking di cop'es of integer i for i = I, 2, ..., n + t and then randomly permuting these 2tn integers and picking up edges from this sequence as usual. Note that this is essentially how p(F) is generated - the k-th copy of integer i corresponds to the k-th element of W l . ) To prove (2.39b) we define a random variable


m paint Let m'=T(T, Z7,J. e , red if e, is incident with a vertex of degree < k - 1 in The blue-red subgraph of G,,,,is distributed exactiy as G(n, m, k ) and so Gm*E Z7 a s . as n i s monotone. Furthermore Gm,-l n k as 6(G,,-,) I) I ) be sufic,ently small (of course, here Y = X& k 35

Even then, the two boxed items were not obtainable with sullic'ent accuracy and were completed via ( 2 ) . The entries i n Table 3 lor 17= 150 to 45000 were obtained froin (3) using S = X , + X 1+ X 3 . In order to shorten the calculation, only the large ternis in the sums were used, since the binomial coctfic'ents would otherwise take too long to evaluate. The values up to I ? = 20 took about 1 seco:id to compute as dld the values from n=30 to 100; from 150 onwards the computation took 1.7 seconds. 2. Connectivity and minimum degree

We know from results of Erdoc and RCnyi [30], Ivchenko [41] and Bollobas [lo] that in a certain range o f p = p ( / i )&no\t eveiy G , i s 4nch that the conncctivity is equal to thc niinimuni degree. We shall show now that this I S the cabe in the entiie range ofp.

Theorem 1. Lot 0 < p =p(n) < 1. T/irwii(C,) = J(G,,) j i i r

U.P.

G,.

Proof. We d o know thilt t h ~ s1 5 true -if c (her 1rp(17)ic(G,) then there I S a partition V(G,,)= VI u S u V2 buch that E(V,, V 2 ) = 0 , l S l = d - l a n d 2 < 1 V , ( < < V v 2Hence /. the complement o f G, (which is, of c o u i x , a G,,) contains a K ( r , / i - d + l - r ) for some r , ? - < r < ; ( t ? - d + l ) . The complement of G, has maXlll~~lJl1 degree ti-6- 1 so the theorem follows 11 we show that there are functions I ( / ? ) , d(n)32 such that

(a) almost no G, contains a partition as above with 2 l , and

2

trees isomorphic trees, provided k = l , relation

(12) implies that

Finally, the right-hand-side of (13) isfl(a), and the difference of the two sides is precitely J2(a). 13 Though the series g;vingf, (a) does not converge fast, it is fairly easy to calculate concrete values orf,(c().The series o f J ; ( a )is not too plcasant but it does converge fast. Our next aim is to give bounds forf(a) for all fued values of a. It will be convenient to set g ( a ) = a / ( a ) .

B. Bollobas, A . Thomason

74

Theorem 14. If U, /3 and y are positive reals and u k g (1) = k log c

for every k , k = l , 2, ... . Proof. Let OO, we have

where y=(a-/3)e-Pf'@'. Consequently

if a=yee(@'+/3 and so (14) holds. Set c = e e ( ' ) and u,=l +c+...+ck-'.Then u , = 1 so g(ul)=g(l). Suppose that g ( u k - l ) > ( k - l)g(I) for some k > 2 . Setting p= I, y=c(K-1 and a=yes(@)+/3=ak,

75

Random graphs of small order

from (14) we find that 9 ( M k ) 3 9 ( 1)

as claimed.

+ 9 (arc- 1 ) > k g (11

1

17

The function f l (a)-

a

a-1

Indeed,g(a)>log(a+l)fora l , p=1 and y=----,where C

ee(l , and suppose that

c=6(”=

B. Bollobds, A . Thomason

76

so

+

>log(a+ 1) g (1) - log2. On the other hand, ifg(a)O and all CL in an interval (0, a,) then the best lower bound for g(ao) obtainable from (14) also satisfies g(a,) 2.27 we have

a

Proof. For c! e'/' and b = ~the left-hand-side of(16) is 2 log @

(

2 b ( b - 1)log I +

a =-

--

log c!

For 2.27O, a(n)=o(n/logn) a.e. Gp is such that

P,(G,) =log(a.t tl 1) n + o ( + ) . However, the main advantage of Theorem 17 is that the graph obtained after the deletion of the independent set is still a random graph, so the process can be repeated and we can obtain a lower bound for x,,(G,). Theorem 18. Let p=a(n)/n and suppose a(n) 1 be a constant. Then by the second part oTTheorem 17 a.e. G, is such that thc first colour class found by the greedy algorithm has at least

n l =[‘og(a+’)-crz-/ - -___ a most n , = n - n , CI ii

-

vertices. In the subgraph spanned by the remaining at

vertices the probability of an edge is a

112

n2

ii

a--log(Cc+1)+c - a’

Hence if a.e. G,,,,, satisfies

n2

‘2 2

Random graphs of small order

81

for all &a,-,>...>cc,, arc such that

< 1 5 log 15, -log + c ,f(k, n). The trick lies in choohing a suitable function f. Setting f-0 makes the restricted search the same as the exhaustive search. The function we used is given by

I

where ,f(O, n)=n, p = 1-4 i s the density of the graph and I i s some parameter. The idea of this is that if W,l =f(k, n ) then the mean value for W,l -dw,(u)- 1 -_iLqf(k, n) and the standard deviation is J p q f ( k , n). Variation of the parameter I allows for a greater or lesser thoroughness of search. The performance of this algorithm is illustrated by Table 17; sonie of the entries can be compared with those in Table 16. In both, tables the times are g'ven in seconds. Quite pojsibly other choices off would give better results, but computer time was not ava'lable to allow experimentation. The coloxing algorithm then consisted of repeated applications of the above restricted search algorithm. A slight improvement was obtained by, at each stage,

I

I

Table 17 Large independent sets found in G,,

n I Mean /lo Mean time

150

150

0.6 7.7 0.02

0.4

I Mean /lo Mcan time

250 0.6 10.7 0.10

n 1 Mean flo Mean time

500 0.5 12.3 1.32

I1

99 0.05

250

04 10.7 0.18

500 04 12 5

2.03

using the restricted search 150 0.2 9.9

0.07

150 150 0.0 -0.2 10.1 10.2 0.15 0.20

200

0.0 10 6

- 0.2

0.04

200 0.2 10.7 0.20

0.22

300 02 11.6 1.18

300 300 0.0 - 0.2 11.8 11.9 1.38 2.89

250 02 10.9 0.40

250 0.0 -0.2 11.0 11.0 0.88 1.43

300 0.6 11.1 0.17

300 0.4 11.1 0.42

500 03 13 0 3.95

500 02 13 0 6.32

1000 05 14.0 24.2

1000 0.4 14.1 45.4

1000 1000 0.3 02 14.4 14.4 84.4 127.6

250

200

200 0.4 10.3 0.1 I

200 0.6 10.2

10.8 0.58

Random graphs of small order

89

swapping a vertex in the remaining graph with one in a previous colour class if this was possible and if it decreased the density of the remaining graph. When the remaining graph became sinall the exhaustive algorithm took over fi.0~11 the restricted search to find independent sets; the number of vertices bclow which this was appl ed is denoted i n thc tables by r. Moreover, the very last few classes tended to be very small and it was usually worthwh le finding the exact chromatic number of the remaining graph once it was small enough. The order of the subgraph so coloured is denoted by s i n the tables. (Subgraphs of order 40 could be coloured exactly in a few seconds and it was often quite feasible to colour up to 50.) The value o f 2 used in the restricted search was not a constant, but was allowed to decrease as the remaining graph decreased. This was done by specrly,ng t w o parameters ct and fi, and letting 2 decrease linearly fi-om ct to fi as the order of the remaining graph decreased from 1000 to s. A non-linear variation would probably g've better results but again we were unable to experiment. This more or less conipletes the description of the algorithm. In practice our aim was to get the best colouring we could inside one hour, and to this end we tried three variants of the basic algorithm. A) Choose ct and fi small enough to consume one hour of computer time. B) Bound the size of the independent set to be chosen (say by (13) in the hope that, although this may not be the largest such set, the large number of sets of this size would oKer an opportunity to choose one which would greatly decrease the density of the rema:ning graph. C) Compare, at each stage, the size of the independent set found with the size one would expect to find in a graph of that order and density. If the expected size is greater, make another tiioi'c thorough search using a smallcr value of 1 in the hope OF finding a larger set. The value of d is computed by replacing u and by smaller parameters y and 6. Table 18 Colourings of the graphs of ordcr 1000, with a=O.5, y=0.2, S = -0.5, ~ ~ 2 3 0

a= -0.1,

(For an explanation of these parameters, see the text.) Graph number s

Time taken Colours u x d Graph number s

Time taken Colours used

I 36 48:38 87

2 34 51:56 86

6 39 43145 87

48 50:16 87

7

3 40 54:58 87

4 34 49:11

8 36

9 40

55:31

561.16

87

87

Mean time: 53:23, mean colours used: 86.9.

87

5

34 55:23 87 10

40 67:.10 87

B. Bollobrfs, A . Thomason

90

Table 19 Six colourings of the same graph of order 1000. The parameter 6 denotes a bound on the size of an independent set chosen

r

S

a

B

230 230 230 230 230 230

38 36 40 39 43 39

0.5 0.5 0.3 0.3 0.4 0.3

-0.11 -0.1 -0.3 -0.3 -0.2 -0.5

6

Y

b 13 13

0.2 0.0

-0.5 -1.0

Time

Colours

25:07 28:40 60: I5 83:43 60:05 11454

88 88 88 88 87 87

Table 20 Quick colourings of ten graphs of order 1000, wilh a= 1.0, 8-0.1 and r = 100 G r J p h number S

Time taken Coloui-s used

1 28 3:07 92

2 25 3:24 90

3 35 2:46 91

4 25 3:22 90

5 27 3:09 91

6 30 3:05 91

7 28 3:13 91

8 25 3107 91

9 36 3:18 90

10 35 3:35 90

Mean time: 3:13, mean colours used: 90.7

The third of these variants appears t o be best. The results of colouring ten graphs this way are presented in Table 18. Some results from all the variants are given in Table 19 for comparison. Finally, Table 20 gives results of colouring ten graphs using large values of I where the aim was to produce a good but quick colouring. The times taken are shown as min:sec.

7. Random regular graphs Let $ ( r z , r-reg) be the set of r-regular graphs with vertex set V= { I , 2 , ..., n>. Turn %(n, r-reg) into a probability space by giving all members the same probability. A point of this space is a raitdoionz r-regulargraph oj'order ti and is denoted by Gr-reg. Properties of are considerably harder to study than those of G M and G,, mojtly because the set % ( ! I , r-reg) cannot be constructed nearly as simply as, say, 9'(rz, M ) . Bender and Canfield [4] gave an asymptotic formula for IY(n, /.-reg)/ as n+m. Bollobis [9] gave a simpler proof by using inany probabilistic ideas. The method also enables one to generate regular graphs rather easily. Several properties of the space % ( / I ,r-reg) have been studied by Worjnald [831, ~ 4 1 . Two important questions about regular graphs concern the minimal diameter of an r-regular graph of order N (see Ber-mond and Bollobis [ 6 ] ) . It is not inconceivable that random regular graphs can be used to tackle both questions.

Random graphs of small order

91

i n fact, for a fixed r and large n the best upper bound for the first function was obtained by Bollobris and de la Vega [IS], with the aid of random graphs. For the sake of simplicity we state a rather crude form of this result.

Theorem 20. Let r > 3 be fixed and rn even. Then, as n+ m, a.e. log,-,(nlogn)-log,-,

6r r-2

--

is such that

-1

+

6 diam (Grhreg) 2 and n> k + r , where s(n, i) denote the Stirling number of the Prst kind.

M . Dondajewski, P. Kirschenhofer, J. Szymahski

102

Proof. By (2) it is easy to see that Gk(x,y) fulfils the following recursive equation

for k 3 1. Since

c s ( n , k )(-- n !

1 -lnk(l-x)= k!

X)"

m

n=k

and

c x" m

(l-x)-I=

n=O

we are able t o write (9) in the following form:

Now using the definition of G,(x,y) and comparing coefficients of the series in (lo), one can easily get the relations (8) and

A,(n+l,k+l)=

Is(i-k)l i=k

--

i!

A,(n+ 1, k ) .

To prove (7) we have to show that

holds. It is easy to see, however, that both sides of (12) are equal to Is(n+ 1 k + n!, which completes the proof. 0 Now we are ready to prove the main theorem of this paper

Theorem 2.

Vertex-degrees in strata of a random recursive tree

103

Proof. To prove the above result it is sufficient to show that the right-hand-side of (13) fulfils the recursive relations (7) and (8) with the respective boundary condition. In fact, A,-,(n+l, k+l)-A,(n+l,

k+1)

-(-'I"( -~ (- 1)"+'-' (L)s(n, k - t r - 1) + n!

(- l)'s(n, i) i=k+r

(( -

i + ; I)-(

i; r)))

Similarly, one can see that the right-hand-side of (13) fulfils (7). Moreover, it is easy to see that the boundary condition

is also fulfilled, which completes the proof.

Corollary 2. For arbitrary k, r and n

0

M. Dondajewski, P . Kirschenhofer, J. Szyrnariski

104

Proof. By Vandermonde's convolution in the form

=)';i(

j=o

(-r)(

ji )

k-j

it follows immediately from ( 1 3 ) that

By the application of the following identity (see (12))

we arrive at the thesis.

0

Note that (14) is much better suited for the asymptotics of A , @ , k) than (13) since the bounds of suinmation do not depend on ti. Theorem 3. If r and k are fixed and n-+03 then for k 2 2

(In n)k Ar(n 1, k + 1 ) = - - + ( y k!

+

(In n l k - l i.) ____ ( k - l)!

Furthermore, Ar(n+1,2)=lnn+y-r+O and

A,(n+1,1)=1+0 where y=0.5772

("":-I ~~

... is the Euler constant.

,

+ 0 ((ln ~ i ) ~ - ~ ) .

Vertex-degrees in strata of

Q

random recurfive tree

104

Proof. To prove the asymptotic formulas for A,(n, k ) we have to deal with an asymptotics of Is(n+ 1 , j + l)l/tz! for fixed j and n-+co. It is known that

I1

where Yj are the Bell polynomials and l,(s)=

1 i-',

(see [I, p. 2171). But

i= 1

?(XI

, ... , X i ) = x i

+( ;)

+

x ~ - z x z .. .

and j Pr{s(i)=2k, Z(i)=2j)=k!(k-2)!

1

P ( S , ) Q ( S , ) R ( S , , Q)

slcV,-(i) SZCV,

and calculating the marginal distributions we arrive at the thesis.

0

J. Jaworskf

146

Remark. One can note that, as in [4],we have the following equation E(l(i))=l+tE(q).

Consider finally the distributions of the numbers of predecessors of a given vertex i E V,.

Theorem6.Fork=l,2 ,..., K + L ,

ie V1

Proof. Notice that p(i)=

C

I(j, i ) + l

j s V- (i)

Hence the formula for the expectation E(p(i)) follows by Lemma I . Denote by BISI, S,] the event T ( S E - { i } ) n S2 = 0

T ( S 5 )n(S, u {i})=8

T ( S , )n Ss =0

T(S,) n( S f - { i } ) = S

for S1c V,- (i}, S, c v,.

A random bipartite mapping

147

Then

Replacing P, and Q, by P,/R(S,, P ) and Q j / R ( S lu { i } , Q), respectively, for S, and j E S1 u {i] and using Lemma 3, one can check that

IE

Using the joint distribution of p l ( i ) and p,(i), we arrive at the thesis.

0

Remarks. We have omitted ;I Formula for Var ( p ( i ) ) since it is long and complicated. For the same reason we d o not consider the f.dlowing random variables: k(i)-the number of vertices i n the connected component to which i belongs, p (Al, A,) - the number of predecessors of vertices from A , u A 2 , where A , c V 1 and A 2 c V,. In fact, distributions of these random variables come directly from the basic lemmas, Theorem 1 and the method of the proof of Theorem 6. Finally, note that for i E V2 distributions of s(i), f(i) and p ( i ) can be obtained in the same way as in Theorems 4, 5 and 6 .

4. Special case: P j E 1/L, Q i= 1/K

All formulas which were obtained in the previous section can be considered as functions of two vectors Q = ( Q l , Q 2 . ..., QK)and P = ( P K + , ,P K + 2 ,..., PK+& Tt is straightforward to verify (see [ I , 51) that, for example, Pr {C=l), -E(C), -E(q) and -Ek(C,) are Shur's convex functions of vectors Q and P,separately. Hence, the probability that the graph G,. is connected is minimized and expected number of cycles, cycles of the length 21 as well as cyclical vertices are rnaximized for P, 1/L and Q l I/K. This is the reason why the uniform case (2'; I/L, 1/K) is considered. Let Pj= I/L, Q1= 1/K and K 6 L . Then the following corollaries can be obtained directly from Theorems 1-6.

J. Jaworski

148

Corollary 1.

where by (n), w e mean n!/(n-j)! Corollary 2.

Corollary 3.

i = l , 2 , ... , K

Var(q)=8

1 (K),(L)i i-2E(g)-E2(q). Ki L'

i=l

I t is obvious that in the uniform case (T; l/L, I/K), s(i), l(i), p(i), pl(i) a d P d i ) are not dependent on the choice of ifrom V,. Therefore, wc shall use notation $[I], /[I], p [ l ] , p,[I] and p,[I]. Similarly, for i E s(i), /(i), p(i), p , ( i ) and p d i ) will be denoled by s[2],1[2], p[2],pl[2] and p L [ 2 ] ,respectively.

v2,

Corollary 4.

149

A random bipartite mapping

Pr{s[1]=21c+1)=--

E(s[1])=1+2

( KK )kk ( L )Lk k

L(l-

_-

c -- 1 can be Irealed as a random mapping with restrictions on independence of choices of images in T*.

158

J. Jaworski

References [l] G . H. Hardy, J. E. Littlewood and G . Polye, Inequalities (Cambridge Univ. Press, Cambridge, MA, 1952). [2] B. Harris, Probability distribution related to random mappings, Ann. Math. Statist. 31 (1960) 1045-1062. [3] J. Jaworski, On the connectedness of a random bipartite mapping, Graph Theory, tagbw 1981, Lecture Notes in Math. No. 1018 (1983) 69-74. [4] J . Jaworski, On a random mapping ( T , P,),J. Appl. Prob. 21 (1984) 186-191. [ 5 ] S. M. Ross, A random graph, J. Appl. Prob. 16 (1981) 309-316. [6] V. N. Sachkov, Combinatorial Methods in Discrete Mathematics (Nauka, Moscow, 1977) (in Russian). [7] V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis (Nauka, Moscow, 1978) (in Russian). [8] V. E. Stepanov, Limit distribution of certain characteristics of random mappings, Theory Prob. Appl. 14 (1969) 612-626.

Annals of Discrete Mathematics 28 (19851 159 - 170 0Elsevier Science Publishers B. V. (North-Holland)

PROBABILISTlC INEQUALITIES FROM EXTREMAL GRAPH RESULTS (A SURVEY) G . 0. H. KATONA Matheniatical Institirte of the Hiingoriati Academy of Scietices, H-I053 Bidapest, lI14n~~~ar.v

The aim of the papcr is lo survey the probabilistic inequalities proved by the method based on extremal combii~atorialtheorems.

1. Introduction

To illustrate the main idea of the field surveyed in the present paper, let sketch the proof of the following theorem:

US

Theorem 1. [ 5 ] If 5 and q are independent icleritically distributed randoni variables taking values from a Hilbert-space X,then

where

11 11 is the nornz of X.

Proof. 1. We start with stating the following special case of the Tursin theorem [17]: If a simple graph with n vertices contains no empty triangle (=for any 3 differ-

ent vertices there is at least one edge) then the graph has at least edges. 2. We need the following simple statement from geometry: If a , , a2,a3 E X are of norm >x ( 3 0 ) then Ila,+a,l[ 3 x holds for a pair 1 < i <j63. The three vectors span a 3-dimensional Eucl'dean space. It is easy to see that the angle between a, and a j is < 120" for some 1 < i < j < 3 . Now it is easy to verify in the plane determined by them that ( l a , + a j ( ( 3 x . 159

G. 0.H. Kafona

160

3. The following trivial inequality will be used: P ( 1 1 5 + r l l l ~ X ) ~ P ( 1 1 5 + S l l ~ X 1, 1 t 1 1 2 . 9

llylll>x).

(1)

Suppose for a while that 5 (and 7) can have only m values, with equal probabilities: 1 P ( 5 = a i ) = - (1GiGrn). Let a, be ordered in the following way: Ilalll>x, ...,

/la,,llkx,

m

Ila,,+lII<x,

..., Ila,.II<x. Consider the following graph G. Let a l , ..., a,

be the vertices of G. Two vertices of G are connected with an edge iff the norm of their sum is ax. Then

P(llt+ylll>x,

~ ~ tI(rll/~>x) ~ > ~ ,

= m V 2(thenumberofpairsa,,a,(I 2 x 3 x. The graph G has no empty triangle by Section 2 of the proof. Applying the Turrin theorem for G, we obtain a lower estimate for (2):

Theorem 1 is proved for this special case. 4. To prove the general case two approaches offer themselves: a) Having an arbitrary distribution for let us approximate it with the discrete distributions used in Section 3. This method was applied in [ 5 ] but the roughness of the elaboration led to unnecessary conditions for the distribution of (. Later Sidorenko [I61 worked out this method properly. We do not treat it here in detail. b) The other method can be found in [6] and [7]. Suppose that the distribution of ( is arbitrary. Let the vertex-set X o f G consist of the vectors satisfying l ( o l ( 3 x . Two vertices, n and b are connected if ))a+bll>zc. G is, in general, an infinite graph and it contains no empty triangle. The right-hand side of (1) is the measure, i n a certain sense, of the set of edges of G. Namely, take the direct product of X with itself. Any edge (n,b) means two elements in the direct product X z : the pairs (a, 6) and (b, a). The set of cdgcs is consequently a symmetric set in X 2 . The measure P on X determines a product measure on X 2 . The right-hand side of(1) is the measure of tiic above symmetric set according to this product measure. We have to give a good lower estimate of the measure of this set by terms of P(11(11>x) (the measure of X) under the condition that G contains no empty triangle.

Probabilistic inequalities from extremal graph results

161

If Xhas finitely many, n elements, let the measure of each element be equal to 1. Then the Turan theorem says that the measure of the edge-set ( = 2 (the number of unoriented edges)+)i) is ,< $ 2 that , is, the half of the measure of X 2 . We may cxpect the same stalement for the general case. We will call the generalization of a discrete coinbinatorial statement for the product measures its "continuous version." Later we will precisely show that there is a transition (under very general condition) for continuous versions. Accepting the veracity of this statement, Theorem 1 follows easily by (1). 0

P

The above sketch of the proof can be illustrated with the diagram in Fig. 1 . The aim of the present paper is to survey the results proved by this method.

2. Continuous versions of extremal theorems in combinatorics The next lemma shows the connection between the continuous and finite graphs. Before stating it, let us give some definitions. Let M = ( X , 0,p ) be a measure space, where a is a a-algebra on X and p is a finite measure defined on a. M 2 is the product of M with itself, that is, M 2 = ( X 2 ,02, p2),where a2 is induced by the products of the members of 0 and p 2 is the product measure. If E c X ' is measurable, that is, E E a2 then G = ( X , E ) is called a (directed) graph. Let Y be a subset of X , then G, denotes the graph induced by Y in G, that is, G,=( Y, Ey)

G. 0.H.Katana

162

where E y = { ( a , b ) : a , b E Y ,(a,b)EE). M or p is atomless i f for any A o w P ) < P (4.

E 6,p

(A)>O there is a B c A , B E cr satisfying

Lemma 1. Let G = ( X , E ) be a graph on the atomless measure space M = ( X , Suppose that

6,p ) .

I

holds for any Y satisfying YI >no. Then

Proof. Introduce the notation M"=(X", a,,, p,,)generalizing the case n=2. On the other hand, define

This function is obviously measurable since E is measurable. Take the integral

when a f b . If a=b we use

Probabilistic inequalities from extremal graph results

163

Summing up (3) and (4) for all pairs 1 n o ), assumption [ E { ~ ,..,

(5) and (6) imply

If n 4 c o this leads to p2(E)kcp(X)’.

0

Let 9 be an arbitrary class of graphs G=(X, E ) determined on the measure space M = ( X , CJ, p). 3 is called hereditary if G E 3 implies G, E 3 for any measurable Y c X . Then

can be considered as the continuous analogue of the “minimum number” of edges in 3. Analogously, let us define H ( n ,Y)=min IEl n”

(7)

where the minimum runs over all members of 3 ‘ having exactly n vertices. It is proved in [7] that (7) has a limit if n-co. The inequality

H ( M , 3 ) k l i m H ( n , 3)

164

G. 0.I€. Katona

is an easy consequence of Lemma 1 if M is atomless. However, this inequality holds for measures with atoins supposing that 9 has certain properties. Let G=({x, xI,...}, E ) be a graph, and define GX=({x‘,x”, xl,...}, Ex),where E x consists of the pairs obtained by subst.ituting x either by X’ or S’‘ i n any way iii any pair which is in E. I n other words, we form two copies o f x i n all edges of G. 9 is called doublable iff G E 9 implies C” E 9 for any vertex x of C.

Theorem 2. [7] Suppose that 3 is a hereditary class of gruphs on flic measure space M ,

if M is atoniless or 3 is cloicbkuble.

For our applications we need this direction of the inequality. One may guess, however, that equality holds in (8) under some reasonable conditions. Indeed, if 3 is doublable then (8) holds with equality (see [7]). However, there is another class of g’s, for which the equality in (8) is proved. Y is called strongly hereditary i f (i) 9 is hereditary, ( i i ) adding a new edge to a member of 3, the new graph is also in 9, (iii) adding a new vertex to a member of 9 (until a certain fixed cardinality) with all the possible edges containing x, the new graph is also in 3.11 is proved in [ I I ] that to any strongly hereditary class 9 there is another class $9” of graphs that a graph H has all its induced subgraphs froin S iff the complement fi contains no subgraph from 9”. gois called in the literature the class of forbidden graphs. The equality in (8) for strongly hereditary graphs is an easy consequence of a theorem of Brown, Erdos and Siinonovits [3]. (The conditions of this theorem and of Theorem 2 are stated incorrectly in

171.) The above results are formulated for directed graphs but, in fact, we need them for undirected graphs. The connection is obvious: each edge (a, b) ( a # b ) of an undirected graph is replaced by two oppositely directed edges (a, b), (6, u). Let us remark that Bollobas [2] independently proved (8) with equality for strongly hereditary classes of undirected graphs o n an atomless measure space. His proof is easier for this special case. Let us see how we can obtain the “continuous version” of the Turin theorem by Theorem 2. Let Q be the class of all graphs G=(X, E ) such that (i) (a, b) E E iff (b, a) E E, (ii) (a, a) E E for all a E X, (iii) if a, b, c are different vertices (E X) then at least one of (a, b), (b, c), (c, a) is in E. By the usual Turdn theorem, the graph G=$,

E), IXI=n, G E Q must contain at least

r(n~l)z] ___

pairs of edges

Probabilistic inequalities from extremal graph results

165

(a, b), (b, a) (a#b). Hence, by property (ii), the number of edges is

This implies Iim IT(//,‘3936 and (8) implies f f ( M ,Y)>+. The proof of Thcorcin 1 can be completed if this inequality is used for measure space induced by the set i n thc probability measure P, and for the set

{/I.Y)

is a consequence of H(M, 9)3 4. Let us remark that [7] states the results on the “continuous versiotis” for g-graphs, however the equality in (8) is known for strongly hereditary graphs only when g = 2 . [7] also contains some results for the case when Ad has atom and $4 is not doublable. Finally, [I;] gives the “continuous versions” of a coinpletely different class of combinatorial extremal problems: a transformation Tof g-graphs to h-graphs is given; the number of vertices and g-edges is fixed, the number of edges of the transformed graph has to be minimized.

3. Two random variables One can sce w i t h an easy construction that Theorem 1 is sharp in the following sense. For any p ( O < p < 1) and any s > O thcrc i s a distribution of 5 (and r7) i n a more than two-dimensionaI space \ L I C I ~t h a t ~ ( 1 1 < + i l l l > x ) = j - p 2 andp=P(II;II >,I-). 111 other words, P(ll c sis) used in place or

‘(llrll >.\-I. Theorem 3. ([ I61 and [Y] independently) Let X b e at2 iriJiniie-dime/~siorialHilhertspace, ( and 11 be X-valiiett iriclcpoiclerzt, Meriticdly distributed raiirloin vtiriahlcs, t h z the best possible junctiorzs f in the i/zequality P(IIx)>f(P(llsT112cx)> are the followi~igones:

G . 0.H. Katona

166

{

f(PI= -E2 p (

if ~ 2 3 , -p ) o t h e r w i s e ,

. 2 p - p 2 if p > + , otherwise,

f (P)={p‘”

f(P)=3P2

when + < c < 3

w ~ i e n
when l < c < JS 2-

>

Each row of the theorein can be proved following the proof of Theorem 1. that is, the scheme g’ven in Fig. 1. We show a new phenomenon of the proof in the case J3/2 d c < 312. We start with a very brief sketch of the proof. Fix the real number x>O and i - x } , X z = X - X 1 . The graph G = ( X , E ) is defined put X , = { a : L I E X ,Ilall< 4s by E={(a, b) : Ila+bll>x}. The following simple geometric statement is true. J 5-

J5

If a,,a,,a, are vectors in a H’lbert-space and ~ ~ ~ z , ~ Ila,ll> ~ > 2 -- ~ then there is apair ifjsalisfying ( ( a , + a , i ( (I3. Hence thegraph G has noernpty triangle with at least two vertices in X,.If G‘ i s finite and IX,=n,, lXzl = n 2 then according to Lemma 2 of [6] the number of edges is at least

I

if n 2 > n , and

(’i2)

the statement for

otherwise. Thc “continuous version” of this Iunma proves

J5 -2

.x) in terms P ( f , ( < ) > e x ) . Examples are,fl(1, 11u21131, llojll > I , J(u4I/> I implies that there are 3 distinct ones of t~iemso that llui +a, +a,,[[3 1. However, tlicre is a little trouble with the combinatorics. We need the ~ninirnumnumber T(t1,4, 3) of 3-elcment subset of an n-element set under the condition that any 4-element subset conlains one or thein. It is conjectured that 4 ? ’ ( I ] , 4,3),( -- . The proof of this conjecture would imply 9

+([/I

;)+

11( 1 +c211.

G. 0.H . Katona

168

for any 3 independent, identically distributed random variables i n a Hilbert-space (see [ 121). The first problem I S that even the order of magnitude of this eilimate is not correct. It is proved i n [lo] that

holds if P ( \ l ( , l l > x ) < : . (10) IS 111iidi stronger for \ J l l d l values of P(ll;ll[ax) than (9). However, the constant i i s not the be,t possible. The reason why the situatuon here I F d IrcIent fiorii the case 1=2 i s that the srnall vectors alw play role. This piobleni is circumvented if we considel P(llcx). Finally, let us mention another reuilt of Sidorenko [16]. He gives lower estiinates of

~(llc]

YI'

IE~~

s (I:

- 3 s ) < e- p s (n - 3s)

6. Other pairs of vertices from V are adjacent with probability p and nonadjacent with probability 1 - p . It follows froin ( I ) and 1-6 that the probability of getting a random graph from %:(n) is less than

Since 1 p = --- (Icg n +log log I 1 n-1

+ q ( n ) ) n/4. By Lemma 3, the probability of this event tends to 1 as n+m.

A new version ofthe solution of Erdiss and Rhnyi’sproblem

177

Consider s=Ln/4] paths with distinct final vertices which one can get from TG by permissible transformations. We denote by V , the set of these final vertices. 3n If there is in T, less than n vertices then, according to (2), with probability log I1 tending to 1 as n+w, there is a vertex in V , adjacent with a vertex outside T G . This contradicts the stability of the path T,;. 0

Proof of Theorem 2. It consists of two plrts. First, we will prove that, with probability tending to 1 as tz+Tx), there i s in a random graph from 9,,(rz)a hamiltonian path, i.e. a path crossing each vertex once. Part 1. Let 11sdenoic by 3;(1z)the set of these random graphs C E 8,,(tz) in which 3n tl vertices and (If(TG)( > -. By Lemmas every stable path TGcontains at least n - ~log n 4 2 4 it follows that, with probability tending to 1 as n-+co, a random graph from ? I p ( 1 7 ) belongs to the set $ ( I ? ) .

Z.!

, n’=n-ka nd 11 represented i n the form Put k = 2

V’={u,, ..., unr}.Iti s e a s y t o s e e t h a t p c a n b e

1 p = --n’(Iogn’+logIogd+ p’(n’>)/ 2 where cp’(n’)-twas 12-03. Thus we can use Lemmas 3 and 4 and therefore, with probability tending to 1 as I I - + C C ) , a random graph from gP(n’)belongs to the set 9 (n Let G be a graph from 9i(n’),7,- a stable path in G of maximal length, V1 the set of vertices lying on T, and IV,I=I. The vertices outside V1 are called periplzcral. According to Lemma 4, the number 1 fulfils the inequality I).

I > n‘-3n’/logn’>

11‘-

3nllogn.

} { u , , - ~ + .. ., v,,), V= 1’’ u V 2 . Here we describe Put V , = { u , ~ , + ...,u , ~= such a process of generating random edges between the sets V 2 and V’ and at the same time of lengthening the path T,, that allows LIS to conclude i n a 4mple way that with probab Iity tending to 1 a> 1 7 4 T I , theie is i n a random graph from %,,(/J) a haniiltoninn path. These edges are genet aled ~n a few steps. Now we present the first one. For each veitex U,E H(T,,) we c hoox froin the \et sZ(T,) a path with u, a\ the final vertex. Then we take a vertex from H(T,,) w i t h the minimal index (my u r l ) and jo:n by an edge every pair of veitices u , , , pi,, j = d + I , ... , t i , with probability p . Theae pair\ ol‘vertices we call utilized. If no edge has appeared then 111 a

173

A.

D.Korshunov

similar way we are looking at pairs u 1 2 ,0,. where ulz is the next vertex rrom H(T,). This process is continuing until there appears at least one edge. Suppose ( v l l , u,,), u j , E V 2 , u l , E H(T,,) is such an edge. If all vertices of II(T,) are utilized and no edge has appeared then tlie process is terminated. If the edge ( u j l , vi,) has appeared then we consider a path with vjI as the initial vcrtcx, consisting of the edge ( P , ~ u, i l ) and that path froin d ( T G )which has u L I ;IS tlie linal vertex. We lengthen this path to a stable one in C and denote it by T‘. We assoiate with T‘ the set d(T’) consisting of paths with the initial vertex u , , , obtainable from T‘ by a sequence of permissible transformations. Every unutilized p:iir I , , , , I ’ ~ where , cil E P”, we join by an edge with probabilily p and call it utilized. 11’ at Icist one edge ( v j l , vi) appears, where u, E H(T‘), then we get a cycle consisting of the edge ( o , ~ , , rind a path from d(T‘)which h;is ui as the final vertcx. It’ no cycle has ~ ~ P C ~ I ~ Cbut LI d ( u j , ) 3 2 then we include u,,, into the set ofperiplicrd vertices. Otherwise M’C stop the process of lengthening tlie p:ith T(;. Let a cycle appear. Since G is corincctcd then there is a vertex u’among pcriphera1 vertices, which is adjacent wilh ;I vertex from this cycle (say u”). Let u“ be adjacent to vertices uiand u, on the cycle, i is measurable, and we call its probability thepercol~itionprobubilitj,.Since the number of black edges froxi any particular vertex is assumed to be finite almost certainly, the percolation probability also indicates the probability that an infinite number of vertices can be reached from the source vertices along black paths. Let Z, denote the set of all infinite paths froiii the source vertices. The percolation probability can be expressed by means or Z, : Lemma 3.1. The percolation probability is equal to P(%"""). Proof. If #a(m) is infinite then either (0 contains an infinite path from S , i.e. 0 E a',, or w has an infinite number of edges from some vertex. S nce the latter event has probability zero, the set {LO : #a(to)=co}\d'" is a null set. But since a?+@-c{w

:

#D(o)=o3}

we have

P ( { w : #a(w)=CO})=P(%+).

0

The following theorem, a version of the "General Clutter Percolation Theorem" of McD armid [8] is very useful Tor comparisons of percolation probabilities on random graphs with different probability measures.

Theorem 3.2. Let (G, P ) and (G, Q) be two locally dependent random graphs, dcjined on tile siinie directed graph G, with nvoitkince furictions { p " ) atid ( ( I , ) , respectively. If p. >, for ewry v E V IVC Iiuuc p(%') < Q> has a product representation $ und o d y if the following titlo ronilitions are vaiid for every u E V: (i) there exists u$nite sirbset E o u cE,, s t d t that d,(K)>O $arid only if E o v cK ,

and (ii) n,,(K)=

n d,(J u Eo")

(- I ) #

(K\J)+l

dl

J c K

for every finite

cind

rionctnpty K c Eu\Eou.

Note that if r/,(0)>0 then E , , 01' condition is the empty set. For more deta Is of the product repiesxtation and lor a pioof of Theoieiii 4.1 see Kuulasniaa 161. (q)

5. A spatial general epidemic model Molliaon ",I has delincd ;I spatial general cp'dcmic G E ( Z d ,CI, 1 1 , F ) as follows. Let [lie set ol s i ~ e sbe z', t ~ i eAIimciisioiwI inccger Iatlicc, ailti let s bc a liiiite subsct or Z d .Wc ;issiiIne lhat c( is a s l r clly positive real numhcr, l r is ;I probab!l,ty density dcfincd o:>Z dsuch t h a t p(O)=O and F i s a piobability d.siribut oil t'unctioii concentrated on (0, cr-1). At time zeio lhei-e is ;in inlect.ous ind,v dual at each site oi's, and the reht oi'the sites ai-e ozct~ped by healthy individuals. Thc inf'ectives emit g c r ~ ~independently ia in Po.s:on piocc.sses with rates CI until they ate reniovcd, each independently arrer having been infectious i'or a random leng~liot time with distribution F. After an indlvldual has becn removed, her site r-enia.nsempty for ever. Each emitted germ goes independently to a site whobe location with respect to the location of the parent is choben according to the contczcf distrSu/ion

K . Kirrrlasmaa

186

p . If a healthy individual gets a germ she becomes infected and starts to emit germs until she is removed after an infectious time with distribution F. If an infected individual or an empty site receives a germ nothing happens. Mollison [9, 101 has studied tlie velocity of tlie front of tlie corresponding simple epidemic, where the inlkcted individuals remain infectious for evcr. His results provide upper bounds also for the velocity of the general epidemic. The most important question about the general epidemic is whether it I s possible that tlie infection never dies out. This happcns alniost surely if and only if infinitcly many individuals arc iiltiniatcly inl'cctcct. Indeed, we ca11make a simple comparison to lind out that the process never explodes, or in other words, that only LI finite number of individuals will become infected in a linitc time: The number of infections in a general epidemic is dominated by the coi-responding Yule pi~ocess, where every emitted germ c:iuscs ;I iiew infection and the infectivcs remain infectious for evcr. The pidxibility of explosion of a Yule process is zero (Fcllcr [ I , Sections XVlI 3 and 41). The problem oF exlinclion of the epidemic rctiuces to one of' I-andorn graphs. Let G'=(v,E ) be tIie simple p p I i where V=Z" and for each u E V , E contains an edge fro~iiu to w if and only if p ( ~ o - u ) > O . Corresponding to the general epidemic CE( a , p , F ) , we define a locally dependent r;indom graph (G, P ) such that, assuming tliere is :in iiilictive at every vertex of V , the edge I'rom u to 10 is black if and only i f tlic infective at u sends a germ to IV before she is removed. The ranctom graph ( G , P ) detcrnmilles the uitimale spread of the epidemic: the individual at u E V, u 4 S, will sooiler o r laler be infected if and only if i n tlie random graph tliere is a black path from S to u. The percolation probability PC6'+') (see Leinnia 3.1) indicates the probability that the infection never becomes extinct. Tlieore~ii3.2 and the knowledge about site pcrcolation processes can be used to prove a thresliold theorem lor the general ep:demic process (Kuulasmaa [ S ] ) . I t states that if i n tlie general ep-dcmic G E ( Z d ,a,p , F), (122 and 11 is properly at least two-dimensional, then there exists a critical infection rate &, such that for a c?, the probability of extinction is less than one. It is interesting to note that i T p is one-dimensional and has finite mean and if also F has finite nieaii then the probability of extinction is always one (Kelly [3]). Compared with an arbitrary general ep'demic, one with constant lifetime is rcmarkabty simple since tlie colot~rsof the edges of the random graph corrcsponding to a gcncraI epidemic G E ( Z " ,a,, ~ i F, ) , where p > O at at least two sites, are mutually independent i f ~ n otily d i f F is degenerate. A proof of t h i s statement is included a t the end ol' this sect.on. Let CE(Z", c?, 11, F ) be an arbitrary general epidemic with (C, P) as the corresponding random graph. Wc can definc two constant lifetime cp'deiiiics such that in the random graph, ( G , P*) say, OF one of them the marginal probability for

v,

Lordly itepen~lentratrdonr graphs

iir

epidemic tnodels

187

any edge to be black 1 5 the came as i n (C, P),and in the other, which has contact dijtributron ji, the probability that an itiTecti\e emits no germs I > the same as in G E ( Z d ,a , p , F). Let (G, P o ) be the random graph of the latter constant lifetijne proccj\. w e can u\c Thco:eni 3.2 to find out (Kuulasmua [5], Kuulauiiaa and Zachary [7]) thdt theic co i 5 t a n t ~ i f e t l m eep tlemics provide both an upper hound and a lower bound for the probabilily of no extinction of G E ( Z d ,a,1 1 , F):

The randoin graph ( G , 1”) of GE(Zd,a,p , F ) h a s ;I product representation if .. (.- ~)‘-‘t//‘’’(.y)>O i n tIic i n t e r v a l O O and p(iv)>O and let F be 110iitl~gu11cr:itc. If T 1s a r:ltidoin variable with distribution F we ciiti make use 01‘ the ineq~~;ihly above to see that

188

K.Kuulasmaa

where p ( { o , w}) is the probability that the edges from the origin to u 2nd w are both whttc and p ( { u } ) and p ( ( w } ) are the corresponding marg~iialprobabilities for the edges. IHence the colours of these two edges are not Independent.

References [l] W. Feller, An Introduction to Probability Theory and 11s Applications, Vol. I, 3rd ed. (Wiley, New York, 1968). [2] J. M. Haminersky, Comparison of atom and bond percolation processes, J. Math. I’hys. 2 (1961) 728-733. [3] F. P. Kelly, In discussion of Mollison [9] (1977) 318-319. [4] C . G . Khatri, On certain inequalities for normal distributions and their applications to simultaneous confidence bounds, Ann. Math. Statist. 38 (1967) 1853-1867. [5] K. Kuulasmaa, The spatial general epidcmic and locally dependent random graphs, J. Appl. Prob. 19 (1982) 745-758. [6] K. Kuulasniaa, The product representation of a locally dependent random graph, Stochastic Processes Appl. 17 (1984) 147-158. [7] K. Kuulasmaa and S . Zachary, On spatial general epidcmics and bond percolation processes, J . Appl. Prob. 21 (1984) 91 1-914. [S] C. McDiarmid, General percolation and random graphs, Adv. Appl. Prob. 13 (1981) 40-60. [9] D. Mollison, Spatial contact models for ccologicnl and epidemic spread, J. I 3 log n vertices.) Dcnote by CY, the set of these vertices (i.e. which form a complete graph i n (W, F , ) of size f , (n)) and consider the graph ( W , , [W,12 n Fz). This graph again does not contain H as an induced subgraph and hence contains a complete subgraph withf,(n) vertices. Iterating this procedure, we get a subset W!E W withf; ( H ) vertices which form a complete subgraph in G. Hence, according to (I), we have

1 log [4(2t+l)log(31ogn)] >1 loglogn 2

Three remarks on dimensions ofgraphs

205

as this for n>no I loglogn 2 >--(log log log n)2

which contradicts (2). In [I] Erdos and Hajnal state also the following.

Theorem 2.4. For every positive integer t there exists E>O and ntl= n , ( t ) such that for every graph H with t vertices a d every graph G with n >n, ( t ) vertices and with the property that neither G nor its complement contains a complete bipartite graph A , B , /A1= 1B1 =n, the graph G contains H as an induced subgraph. Imitating the above proof of Theorem 2.1 and using Theorem 2.4 instead of 2.3, we obtain a proof of Theorem 2.2.

Corollary 2.5. Fix E>O. Let d be a class of graphs closed on induced subgraphs. Assume that there exists a graph which does not belong to .d. Then there exists a graph C = ( V , E ) which fails to be an induced subgraph of product of less than (1 - E ) log log log V ( graphs belonging to d.Particularly, if G = fl Ai ( A , E d ) ,

I

ie I

then

1'

>(l-&)logloglog n

3. An example of graphs with high dimensions Given a graph G with n vertices, deno'.e by dim(G) the smallest number d such that G is an induced subgraph of the graph

Ki x 7 x K,, =(fQd. L -

d

Here x denotes the direct product of graphs. This concept was defined in [3] and [4]. Clearly, i f d denotes theclassof all complete graphs thendim(G)=dim,(G) for every graph H , where the symbol dim,(C) was introduced in Section 2. It is proved in [3] that dim nK, = [log nl+ 1 and dim(K, +K,)=n, where nK2 is the matching of size n and K,,+K, denotes the complete graph with n vertices together with an isolated vertex. These results are complemented by the followiag.

J. NeSetr'l, V. Rddl

206

Theorem 3.1. For every positive integer n there exists a graph G, with the following properties:

+

( I ) G, is bipartite (and consequently does not contain K3 K,), ( 2 ) G, does not contain 3K2 as an induced subgraph, ( 3 ) dimG,>n. Proof. Let n be fixed. Let G=(V, E ) be a graph which does not contain a cycle of length 2 6 and which has chromatic number at least 2""+1. Denote by G, =( W ,F ) the following graph: W = V x { O , l}

iff i#j, u # u ' and { u , u ' } E E

We prove that the graph G, satisfying the above conditions ( I ) is trivial, (2) follows from the fact that C d o s not contain short cycles: if the disjoint edges ( ( v , , i,), ( u 2 , Q} ( ( v 3 9 i3) ((05

Y

9

( 0 4 Y i4))

is) (vfJ9 i d } 5

of G, form an induced subgraph then it is easy to see that at least 4 of the vertices u l , say v L ,v 2 , v 3 , u4 are d stinct. But then these vertices form a rectangle in G. In order to prove (3) let us first recall the following fact which is made explicitely in [4].If dim H < k then the edges of the complement of the graph H may be coloured by k colours in such a way that the graph formed by the edges of any of the colours is a disjo nt union of complete graphs and each edge is coloured at least once. Thus assume on the contrary that dim G, 3 we have

tz

and

for any fixed y>O. As a matter of fact, in the worst case, when q > p , i=i(n)-+(x, and j=j(n)-+oO as n+cm the left-hand side of (7) is at most ( q / p ) w + o ( l ) = o ( t z y ) , since w=o(log n). Therefore, we dcduce that the right-hand side of inequality (5) is < 1 and consequently

where the summation is over all pairs (i, j ) such that 2 < i + j d r + w and O < S < 1 is a constant. Similarly, one can check that

where 2 < i + j < 2 w . Thus, by (4) and the above facts we have

Prob (X=0) d Var ( X ) / E(X ) 2= o ( 1 1 - -’) , where r and w satisfy (6). This implics that

Bipartite complete induced subgrapks of a random graph

Now, let us choose k = k

(12)

213

such that

Then, by ( 2 ) Prob(X> 1)<E ( x )

(s>

f

6

rw/(r+w) r + w

~

r !w !

(nq(r + w -

)

=o(n-I)

for any integer i. Therefore,

(

i 13aI)

Prob Bn(r,w ) > ( 2 + ~ ) - - -

=o(n-').

(9)

Consequently, by the Borel-Cantelli Lemma we deduce from (8) and (9) convergence of &(r, w ) with probability one. Now, to prove convergence in any mean, let us fix t > l and 0 112. 0

From Theorem 2 one can deduce that when w is of the same order of magnitude as r , then for small as well as large values of the edge probability p , both the lower and upper bounds on Bn(r,w ) are wide apart. The best situation holds when p = 1/2. As a matter of fact, the following sharp result is true. Theorem 3. Let B:(r, w ) stand for Bn(r,w ) uhen considering a random graph G(n, 1/2), Ifw,-cr, when k+co and O < c 6 1 then the sequence (B:(r, w ) } satisjies

with probability one and in any mean.

3. Bipartite cliques For the given two non-decreasing sequences of positive integers r = ( r , , r2, ...) and w=(w,, w,, ...), where r f w is increasing, define amaximal bipartite complete subgraph Krk,wk (a bipartite clique) of a graph G. A complete bipartite subgraph K,,,, of G is maximal if it is not contained in any other subgraph of G being a Of K I k + l r W k + l * Let b,,(r, w)=min{r,+w,:

there is a copy of maximal K,,,, in G(n, PI}.

logn First, we will show that bipartite cliques of order less than (l-&)-are unlogf likely to occur. Remember that f and h stand for the maximum and minimum of two numbers

(-,P --), respectively. We have the following result. 4 1

1

Z . Paika

216

Theorem 4. For every c > 0 log n b,,(r, w ) > ( l - ~ ) - - lo g f

as.

Proof. Let Y= Y,,(k) be the number of all maximal copies of Kr,win G ( n , p ) , where r=rk and w = w , . Let k be fixed and assume that rfw and p < 112. Then

It is clear that the same upper bound can be used in the case when w = r and p 6 1/2. Now let us put t=

[

(1-E)-

12;J

and assume that k=k(n) is such that r k + w k < t . Then for sufficiently large n

Consequently, Prob ( b n ( v ,w )6 [(I

-8)

5f-I) c'

log 1 l P


induces in G a subgraph containing a copy of K. First we formulate a result for a random graph ING,,(,)(n). We will say that for given (ING(n)), (p(n)) and a graph H the condition B(r) holds if

and (ii) for every Sr

b,,(~r)(p(n))"'X"=o((b,(H)(p(n))'>3,

n+m,

r = 2 , 3 , ...

.

Subgraphs of random graphs

223

Moreover, let us denote by B*(r) the condition obtained from B(r) by putting -r instead of r in the right-hand side ol'(ii).

Theorem 1. Let X,, be the number of all H-graphs contained as subgraphs in a random graph INGp(,,)(n),p=b,(H)(p(n))' and n-t co. Then (A) Ifp-.O then P(X,, > 0 )= o( 1); (B) I f p - t c > O and for every r=2, 3, ... the condition B(r) holds. then X,,Po (c/a(H)); (C) I f p - t co and the condition B(2) holds, then P(X,, = 0 )= o (I); (D) Up+ co andfor every r =2, 3, . .. the condition B*(r) holds, then X,,-bN (0, 1).

An analogous theorem can be stated for a random graph ING,,(,,,(n). First, let us start with an appropriate reformulation of the conditions B(r) and B*(r). We will say that for given (ING(n)}, (A(n)} and a given graph H the condition U(r)holds if (i)

ai-lu,(rH)-ui(H)

and (ii) for every &',

u,,(&',)/a,,=o((u,(H)/a,,)'), n+co , r = 2 , 3 ,

... .

Moreover, let U*(r) denote the condition obtained from U(r) by putting -r instead of r in the right-hand side of (ii).

Theorem 1'. Let X,,be the number of all H-graphs contained as subgraphs in a random graph INGA(,,)(n),p=u,,(H)/a,,and n+co. Then /he assertions of Theorem 1 hold if we replace B(r) and B*(r) by U(r) and U*(r), respectively.

Proof. Let us note that EX,=,u/a(H). Thus (A) is trivial because the inequality P(X,#O)<EX, holds. In order to prove (B) we will show that the r-th factorial moment of the random variable X,,, Er= E(X,(X,,- I ) ...(A',,-r+ l)), tends to (c/a(H))' for r= 1,2, ... as n+w. Then the thesis follows from the fact that the Poisson distribution is uniquely determined by its moments. It is easily seen that E, is the expectation of the number of all ordered r-tuples of H-graphs contained in ING,(n). Therefore, we can express E, as E,= E; + Ei', where E: counts all. ordered r-tuples of mutually disjoint H-graphs. Observe that E:=b,(rH)p"/d(H),

r = l , 2,

...'

A . Rucitiski

224

and for every P r

EL’

=

o ( b , ( ~ , . )pe

(xr)),

r =2 , 3 , . .. .

Thus, from the assumptions it follows that Ei--+(c/a(H))’ whereas E:’=o(l) as n-tco, r = 1 , 2, ..., and the proof of (B) is complete. To prove (C) we shall use the inequality P(X,,>O)

> (EX,,)’/E:(X;)

=

( ( E i +E;’)/(EX,,)’f l/Ex,l)-‘.

(1)

One can check that under the assumptions of (C) the right-hand side of (1) tends to 1 as n+co. In order to prove the last statement of Theorem 1 , we have to determine the r-th moment of the random variable

x,,

+cz

=c1

where S(,) is the Stirling number of the second kind. From the condition B*(r) it follows that C2=o(l) whereas X I can be expressed (see [3]) a s (EX,Jke-

=(Var X,J - r ’ 2

k !)- ‘ ( k - EX,,)’

k=O

Since in our case VarX,I-EX,, if EX,,-+co then C, tends to the r-th moment of the standard normal distribution and the theorem follows. The proof of Theorem 1’ is analogous and therefore omitted. One of the possible extensions of Theorems 1 and 1‘ is to count all H-graphs in a random graph such that 11 is an element of a specified family 9 l of graphs with not necessarily fixed order, i.e. with order which depends on n and tends to infinity as n-cn. (For particular models see [2, 9, 10 and 171.) Wc are also able to state similar theorems for some other random structures such as random directed graphs, random niult’graphs, random hypergraphs, random signed graphs as well as for soine other kinds or subgraphs, e.g. induced subgraphs and isolated s u bg r aph s .

Now we turn o u r attention to soine possible applications of o u r results to different random graphs and other random structures. First, notice that the several known facts on subgraphs of complete random graphs K , , p and K,l,N

225

Sitbgraphs of random graphs

( [ 2 , 3 and S]), bipartite random graph Kn.m,p ([lo]), random lattice ([13]), and random graph with given vertex degrees G,,, ([16]), follow immediately from our main result. In this section we present a number of corollaries of Theorem I’ dealing with an out-regular directed randoin graph, a randoin tree and some random covers such as random permutations, partitions, matchings and forests. 1. An out-regular directed random graph Suppose that each of n vertices chooses randomly d neighbours, d> 1, so that all (“d I ) possible outcomes are equiprobable. A resulting directed random graph -+ D,,,dis of the uniform type with ING(n)=K,,, the directed complete graph, and A ( n ) = ( D : d + z d ) , where rl+ means the out-degree sequence of a directed graph

D. Let us notice that

a,,=

(“a1>”

and if F is a directed graph with out-degree

sequence (L/,, .. .,d,,,), di 1 and ( x ) ~ =1. Moreover, any sum of not all pairwise vertex disjoint cycles has more edges than vertices. So, we arrive a t the following corollaries.

Corollary 1. If a directed graph F has more arcs than vertices then P(D,,, contains an F-graph) =o (1) , n-* co

.

Let X, be the number of H-graphs contained in Dn,d, where H is a cycle of the length k having out-degree sequence ( d l , ,..,dk) (o . m i n ( g l , h k } } . So, the bound of summation in ( I ) may be replaced by Jnin ( g I , bli) --s. On the other hand, if we define S,, =

{(

for all

112

€.A'"', this boundmay be set to infinity. How-

ever, we set out tentatively with version (1) using bound b g - . ~ . Consider P ( A I I A , ,... A,,,) with 112 E . N & Except for the parameters b, g , k and 1 this probability depends on two partitions of 111. To see this, regard the undirected graph ( B u C , (il, . . . , i,,,}). The degrees occurring in B form a partition of rn ( p say), as do the degrees of the vertices i n G (constituting i ) .Clearly, if (p, ;)$ A i , x A!,, or ( p ( > b or (g, then P ( A , , ._.A,,,)=O. Furthermore, i f ?,= 1P12P1 ... kpkand ;= 1q'2q2... P',then P ( A , , ... A,,,) equals

4)

For all (p, E A:, x A:, (if the pair is graphical or not) we define weight w ( p , ); by expression (2). Let N(?, i ) be the number of bipartite graphs with respect to B , C> such that arises i n B and in G, then we may state

5

or for short S,,=C(Nw)(p, G). Summarizing, i n order to obtain explicit expressions for P ( X + =s) it suffices to evaluate the N(p, G)'s. Generating functions can be found for N ( p , applying methods given by Read [3]. However, we prefer to follow another approach, which is more straightforward and leads more easily to explicit expressions.

G)

5. Generating polynomial functions

Let F be a function of b+g

+ 1 variables s1,.. ., x,,, y,, ...,y,

F ( x l , . . . , x b , Y l , . . . r y p ; t)'

b

g

i=l

j=1

I1

(I+XiYjt),

and t , defined by

I. H. h i t

234

and let h, be the polynomial coefficient of t"' in the expansion of F. The polynomials h, are invariant under permutation of the indices of the x's. The same holds with respect to the y's. In the natural expansion of h,, each additive term is the product of rn (not necessarily distinct) x's and my's and finally an integral coefficient. If we let x, correspond to b, E B and y, to g j E G, then a term such as cxyx';... x:y:'yy ...$ (Cr,=C.s,=?n of course) corresponds in an obvious way to the set consisting of all bipartite graphs with bipartition ( B , G ) with m lines and such that d(b,);i.e. the degree of vertex b,, equals rl and d(g,)=s,. Furthermore, the coefficient c clearly equals the cardinality of that set. If p E A;, ? E Af, such that IjlGb and l g we drop the conditions lPl66 and (GI < g and (4) holds for all (i, 4) E A; x Af, Using methods very similar to those in the proof of Lemma 3 in Smit [4] h, can be expressed as follows. Proposition 5.1. Let m E ,Nbg. Then,

b

where X,=

C I=1

a

x; and Y,=

C

y;

j= 1

With this result and (4)the following thcorem arises.

.. Theorem 5.2. Let m E Xig,5 E A:, q E A; a i d n=min{k, l } , then

Matchmaking between two collections

n X p ) denotes the coeficient

235

n

where C;(

Similarly C;(

of

n.):Y

s= 1

Proof. If m=O the statement obviously holds. Suppose m E NbS.We have N ( 3 , G)=(!)(!)C;,;(hm). Using Proposition 5.1 and some simple properties P

4

of C;,;, we obtain

m

Since clearly C;(

r[ Xp)=O whenever r is not k-restricted a= I

whenever r 4 A!,,, we may restrict ourselves to partitions So the proof is complete. c]

n m

and C;(

r E A:

Y,'.)=O

a= 1

n A:,,= A:.

In the next section we will show how this theorem can be used to derive explicit expressions for N ( p , < )and so for P ( X + =x), whenever k and I are not too large.

6. Exact results With the aid of Theorem 5.2 we are able to obtain explicit expressions for C;,q(h,) and N ( i , S ) i n the cases k , 163, k G 2 and 1 6 2 . It seems that most other cases are too difficult to deal with i n a similar way. As an example we consider k, 163. In this case, we only have to determine the quantity C-,(X;'X;*X;') for 2; and 1' = 1"2r23'3E A:,. A little reflection shows that C;(X;'X;ZX;3) vanishes whenever p l > r I , p 3 < r 3 , p 2 + p 3 < r 2 + r 3 or 1i1>6. Let ~ ~ = r ~ ~- , =p p~, +, p ~ - r , - r ~ and N 3 = p 3 - r 3 and suppose Ni>O, i= 1 , 2 , 3 , lal.

where rl =m-2r,-3r3,

Q3=min(y3, q3) und

Furthermore,

Proof. By the bounds of summation Q3 and Q z ( r 3 )in the asscrtion, the condition N,>O, i= 1 , 2, 3 for (6) holds for all r2 and r 3 . By Theorem 5.2 the proof can be completed, using the fact that the product

(%)(5)

vanishes whenever

>b

or I l q l > ~ .

For

y,

E

A: we have the following corollary.

Corollary 6.2. Let m E A";,, p atid

E

A:. Then,

Proof. By Theorem 6.1, setting u = r 2 . 0 Obviously it is possible to formulate theorems about the exact probability distribution of X&(l, k ) whenever k 6 3 and 1 6 3 , but this will not be done here. Another application of Theorem 5.2 is about moments of Xf. For example we can derive expressions for S3 and E ( ( X + ) 3 } .

Matrhniaking between two collections

237

7. Asymptotic results

In this section we pay attention to an asymptotic expression for inoments and to results about the asymptolic behavior of X,’(f,k ) ( b and/or g - + c o ) . We begin with a theorem about moments. Theorem 7.1. Let k and I be arbitrary, but j i x d . Then,

Proof. For j = O , 1 this statement obviously holds. Let 2 G j d b g and suppose k 3 2 , 1 3 2 . Then S j = ( N m ) ( l ’ , l’)+(Nu)(l’-’ 2 * , l’)+(Nw)(I’, l’-’ 2’) +X(Nw)(F, i),where the last summation is over all other pairs (p, ( p , ~ ) = O { ( b - ’ + g)-*’>,so S ; “ ’ ~ % ( N 4 ( p4) ,’ O { ( h - ’+g-I)’}. This is even true if /c= 1 or I = 1. If

a

and

then it follows from Corollary 6.2 that

238

I. H. Smir

and from the definition of the Sji)’s it is clear that S,=CSy’ even holds if k= 1 or I = 1. Because

and

the result follows. 0

Theorem 7.2. Let k , 1 and g be arbitrary, but jixed. Then,

x=o,

..., g l ,

where

and

Proof. Since Sf=O whenever j>gI, the upperbound of summation in (1) may be replaced by g f - x. Let rn E Nil and q E A;, then clearly

Furthermore,

Matchmaking between two collections

239.

Since

the proof is complete.

0

As a corollary we state the case I= 1. Corollary 7.3. Let k and g be arbitrary, but fixed. Then,

Proof. By Theorem 7.2, substituting I= 1. 0

For the case that b and g simultaneously tend to infinity we present a further result on the asymptotic behavior of P ( X + =x). Theorem 7.4. Let k and I be arbitrary, but fixed. Then,for x E N' P (X'

=X) = 7~:

{ 1+ C ( - X * + x +2xkI - k212)}+ 0 {( b-

+9- ')'}

,

( b ,g - t w ) , where

':27

denotes the Poisson probability (kl)xe-rl and C=f((bk)-' +(gZ)-').. X!

I. H. Smir

240

Proof. Let T j = n ~ ' { l - ( j ) 2 C and } Rj-Sj-Tj, min (bk, g l } 2x.Then one can show that P(X'=x)=

-y(-l)i i=O

("

EN'.

Furthermore, let n =

+

i, S, + = B1 - B2 B, ,

where

f i) T,

m

( - I);(,

BI = i= 1

+

{ 1 + C ( - x 2 + x + 2k 1 - k 21');

=

,

IU

B2 = i=n-x+

I

(-

I)'(,:

i, T',;,

obviously also a convergent series, and

n -x

B, = 1( - I

i ) R , x ,i .

i=O

For B z , IB,16

(kl)'

1 '3J

x!

i=rt-x+1

,:(lily - (l-(x+i)zC}

holds. Using i ! k ( r t - x + l ) ! x

1.

( i - n + x - I)!, it follows that ~ , = O { ( b - l+g-s)2}. In order to prove that B 3 = O { ( b - ' + g - ' ) 2 } we split up the corresponding R, into 4 parts. Consider the definitions of the S);)'s in the proof of Theorem 7.1. 4

Obviously, Sj=

Sj" holds for all j

E .&;,

even f o r j < I . Let

i= 1

Ri" = Sj('!-

(kl)'

7-r

{I - & ( j ) , ( b -

J-

Then

Rj=

+g -

I)},

1Ry) . i=l

By methods similar to those used in the proof of Theorem 8 in Smit (41, it is possible to show that for i = l , ..., 4 there is a positive function f:B x G - + 9 ? independent o f j and of the order O((6-l + g - ' ) * } , and a positive real number D which is constant with respect t o j , b and g, such that (kl)' (R~'(6,j4(l+D)'j(b, I.

g),

j=O,

..., n.

241

Matchmaking between two collections

Obviously, the same holds for R,. And now it follows easily that B,= O((b-'+g-')2}. !J Finally, we state a corollary which is an immediate consequence of Theorem 7.4. 9

Corollary 7.5. Let k andl be arbitrrrry, butfixed. T i m , A'&(/, k)+ Po (kl), (b,g-' a), where Po (kl) rietiotes the Poisson distribirtioii with parameter kl.

References [ I ] J. Jaworski, On thc conncctedness of a random bipartite mapping, Lccture Notes in Math. 1018 (1983) 69-74. [2] J. Jaworski, A random bipartite mapping, Ann. Discrete Math. (to Ippear). [3] R. C . Read, The enumeration of locally restricted graphs (lI), J. London Math. SOC. 35 (1960) 344-351. [4] I . H. Smit, The distribution of the number of two-cycles in certain kinds of random digraphs, Wiskundig Seininarium, Vrije Universitcit, Amsterdam, 1979, Rapport nr. 116. (51 S . S. Wasscrman, Random directed graph distribution and the triad census in social networks, J. Math. Sociology 5 (1977) 61-86.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 28 (1985) 243-250 C Elsevier Science Publishers B. V. (North-Holland)

FOUR ROADS TO THE RAMSEY FUNCTION Joel SPENCER Department of Mathematics, S U N Y at Stony Brook, Stony Brook, N Y 11794, U . S . A .

In this paper we are concerned with lower bounds to the Ramscy function R(k). We examine four arguments and the bounds they yield. All arguments we consider are variants of the probabilistic method.

The Ramsey function R(k) is defined as the smallest n such that if the edges of K, are two-colored there necessarily exists a monochromatic Kk.In this paper we are concerned with lower bounds to R(k). Thus, for n as large as possible, we wish to two-color K,, so that there does not exist a monochromatic Kk.We examine four arguments and the bounds they yield. Ail arguments we consider are variants of the probabilistic method. In 1946 Paul Erdos [2] published a seminal paper on the probabilistic method. He showed that then

R(k)>n.

A calculation using Stirling’s formula shows that this implies R ( k )>

(--> 4 1

+

k2k/2(1 o (1)).

Here is his argument in modern, i.e. probabilistic, terms. Consider a random coloration of K,,. That is, each edge is colored Red or Blue (our colors) with equal probability and these probabilities are mutually independent. Each k-set has probability 2’-(’)

of being monochromatic. There are

the expected number of non no chromatic k-sets is

(9

such k-sets. Hence

(321-(k)-

Basically, we are here invoking the linearity of expectation. For each k-set A let X, be the indicator random variable for the event “ A is monochromatic.” 243

J. Spencer

244

Let X=ZX,, the summation over all k-sets A . Then X is the number of monochromatic k-sets. The variables X,, XA, can be quite dependent but linearity of expectation does not require independence of the variables. Thus E(X) =CE(X,) which, by assumption, is less than unity. As X is integral valued there is some “point” in the probability space for which X=O. The points of the probability space are the colorings. Thus there is some coloring of K, with no monochromatic Kk,completing the proof. Since 1946 the only improvement on the bound (2) has been in the constant term l/e$. As the best known upper bound for R(k) is of the ordcr ( 4 + 0 ( l ) ) ~one could say that no significant iniprovenient on the Erdos bound has been found. Indeed, the problem of finding the real order of R(k) - e.g. the v:ilue of lim R(k)’lL- is, in this author’s opinion, the most vexing problem involving the probabilistic method. Our three other methods, while not shedding light on this question, do bring basic methodologies of the probabilistic method into sharp focus. An improvement of the Erdos bound is given by the Deletion Method. We show that then R( k) > n (1-&)

.

(3)

Taking E = o (1) appropriately, this implies

(3

R ( k ) > - k2k’2(1+o(1)).

(4)

Again consider a random coloration of K,. As before the expected number Xof monochromatic Kk is (;)2*-(’)

which is now less than cn. Thus there is a particular

coloring of K, with less than En monochromatic Kk. Select one point arbitrarily from each of the monochromatic Kk and delete it from the vertex set. At least n(l -E) vertices still remain. All of the monochromatic Kk have been destroyed so we are left with a coloring of the edges on at least n(I - E ) vertices with no monochromatic Kk. The above argument was discovered by Jim Shearer in 1982. The result is not as good as that obt,tined by the Lovisz Local Lemma (to be described) discovered in 1975 and has not been published, Still, it is surprising that this result, using a fairly well understood technique, was not commented on (to this author’s knowledge) between 1946 and 1975.

Four roads to the Ramsey function

245

Our third argument uses a Recoloring Method. This method was used effectively by Jozsef Beck [ I ] to find bounds on the function m(n) for Property B. We show that 1f

(2>

k

- ( 2 )< ,I 2 - R

21

then R ( k ) > n .

(5)

Here E is an arbitrarily small but fixed positive real and n approaches infinity. This implies

R(k)>

(f) --

k 2 y 1 +o(l)).

Again we begin with a random coloring of K, (which we call the First Coloring) yielding nZ-'monochromatic Kk.For somewhat technical reasons we set s= [ I 0 0 x e-2]2 and call a k-set A nearmono if all but less than s edges are the same color. (Nearmono includes monochromatic.) The expected number of nearmono Kk is then

The additional factor is less than kZS.As k - c In n=no") and c, s are fixed with n approaching infinity this term is l z o ( ' ) . Thus there are n2-e+o(1)nearmono &. Set p = 100/k. G'ven thc First Coloring call an edge critical if it is a red edge in a nearred / 1 and Eq. (20) becomes

Random graph problems in polymer chemistry

259

Eq. (18) yields

In this model Whittle’s results are consistent with the following: as /I increases from zero all components are finite trees until bn> 1, when a component of size qn appears. All cycles, loops and niultiple bonds remain confined to this giant component . These statements are probably only approximately true as more precise statements have not been investigated yet.

4. The tree model

A model for trees, cimilar to those for graphs and pseudomultigraphs using the degiee weights ( H , ) of 4 2 and 0 3 , has not been inve\trgated. Instead standaid modcls begin with the distribution of vertex degrce (I),). p , IS the probability that ;I vertcx has a given degrce (cf. Eq. (12)). Let P ( s ) be the ordinary generating function of {p,].

c PjFIJ. a

Y(n)=

j=O

Because the correspondence between ordered and unordered trees is a function of tree partition alone (see Eq. (ti)), we examine unordered trees as representative of both cases. The probability that the vertex on the end of a random edge has degree k is

kPk f,=-P‘(1)

r

‘I

k = 1 , 2 , 3 , ....

Proof. The probability is the a priori probability of the vertex being a degree k ( p , ) weighted by the number of edges from the vertex (k). l/P’(l) normalizes the (fk) into probabilities. This effectively specifies the distribution of the trees as a branching process (Athreya and Ney [I]), see Fig. 4. Choose a vertex at random. Eq. (22) g’ves the distribution of the degree of this “progenitor” vertex. The progenitor’s degree is the number of I-st generation “offspring” (those vertices adjacent to the progenitor). Each offspring vertex in the I-st generation has degree k (i.e. k - 1 2-nd generation offspring and 1 parent) independently with probability fk, as do offspring vertices in subsequent generations.

J. L. Spouge

260

a

GEN 2

GEN 1

b GEN 0

Fig. 4. (4b) shows the branching process resulting from choosing a random vertex in the tree in ( 4 4 .

Results about tree distributions in this model are derived from the corresponding branching process results. Straightforward extensions of this model are possible: multiple vertex and edge colours, directing, etc. Perhaps the most interesting extension is to assign to the vertices independently identically chosen random masses (a) and to let the vertex degree probabilities (p,) be functions of a. This last notion has obvious extension to the degree weights (H,) of $ 2 and $ 3 . It is likely that an approach to trees through degree weights (If,) would yield results similar to the branching process until a tree containing O(n) of the n vertices formed (this co;rebponds to a supercritical branching process). Thereafter (in certain chemical models), the branching proxss method yields results consistent with those at the end of $ 3 (despite our exclusion of cycles!).

Random graph problems in polynter chemistry

261

Grimmett [I31 has used special branching processes to enumerate trees; the branching process enumeration of trees by partition is central to the tree model of this section (Conclus.on, Spouge [27]).

5. Conclusion This paper attempts to bring Fame coinbinatorial problems suggested by polynier chemistry to the attention of random graph theorists. In $ 2 we ass:gn probabilities to the graphs o1'the Erdos-RCnyi scheme on the basis of graphical partition as well as number of edges. If applied l o pseudomultigraphs (9 3), lhis ncw scheme may produce non-smooih changes In the distribut:on of vertex degrees as the expected numbers of edges is increased. Whittle's analysis, which required novel a~ymp:oticmethods, showed that almost all finite components of a random p;eudornult graph are trees until the threshold for the appearance of a component of O(ti), where n is the number of vertices of the p.,eudomult igraph. In $ 4 we exainine the branching process model for trees and indicate a connection between branching processes and enumerat'on of trees by partition. There is considerable scope for mathematical investigat'on of these models since most of the work on them appears in the literature of polymer chemistry. I g've polymer chemistry references in the Al-,pendix.

Appendix Flory's [5 - 71 RA, model is the paradigm of polymerization models. Stockmayer [24] gave the size distribution for this model. Flvry [8, p. 1921 disagreed wit11 the interpretstion of Stockmsycr's result. Falk and Thomas [.I]resolved Ih: resulting debate by computer sirnulatioil (see also Ziif and Strll [ 2 3 ] ) . I gave analytic results for Flory's modcl (Spouge [29]), not realiring thal i t is the spccial casc

of Whittlc's [I91 pseudoniultigrapli model. Thc Stockmaycr interpretation is equivalent to a trcc model employing jh: samc ctcgrce v,eiglits if/,:. Gordon [ 101 and Good [U] introduccd tllz Branching Process Tree Model and Gordon cf 01. [ I I ] g i v e ii rigor-ous ju.;rilicarion 01' i t \ applicability. Gordon and Scantlebury [I21 ;rnd Spougc. [27] g3vc rclinLm :nts cquivLilcnt to niul~icolouringvcrticcs and edges respectively. Spo~igc[28) illso allow-d the virliccs to h:ivc random mass. Flory's [S, Cli. 91 A, I < t l - g ~iiodclis equivalcnt to employing directed trees. Spouge [25. 26, 281 gives solutions and rctinemcnts for this model.

262

J. L. SpouEe

References [ l ] K. B. Athreya and P. E. Ney, Branching Processes (Springer-Verlag. New York, 1972). [2] G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, Theory and Technique (McGraw-Hill, New York, 1966). [3] P. Erdos and A. Renyi, Math. Inst. Hung. Acad. Sci. Hung. 5A (1960) 17-61. [4] M. Falk and R. E. Thomas, Can. J. Chcni. 52 (1974) 3285. [5] 1’. J. Flory, J. Am. Chcm. Soc. 63 (1941) 3083-3090. [6] P. J. Flory, J. Am. Chcm. Soc. 63 (1941) 3091-3096. [7] P. J. Flory, J. Am. Chcm. Soc. 63 (1941) 3096-3100. [8] P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, New York, 1953). [9] I . J. Good, Proc. R. Soc. Lond. A272 (1963) 54-59. [lo] M. Gordon, Proc. R. SOC.Lond. A268 (1962) 240-259. [I I] M. Gordon and T. G. Parker, Proc. R. Soc. Edinb. A69 (1970/1971) 181-192. [I?] M. Gordon and G. R. Scantlebury, Proc. R. Soc. Lond. A292 (1966) 380-402. [I31 G. R. Grimmett, J. Austral. Math. Soc. A30 (1980) 229-237. [14] G. R. Grimmctt, in: L. Beineke and R. Wilson, eds., Further Selected Topics in Graph Theory (Academic Press, 1983). [I 51 F. Harary, Graph Theory (Addison-Wesley, London, 1969). [I 61 J. K. Percus, Combinatorial Methods (Springer-Verlag, New York, 1971). [I71 P. Whittle, Proc. Camb. Phil. SOC.61 (1965) 475-495. [I81 P. Whittle, Proc. R. SOC.Lond. A285 (1965) 501-519. [I91 P. Whittle, Adv. Appl. Prob. 12 (1980) 94-1 15. [20] P. Whittle, Adv. Appl. Prob. 12 (1980) 116-134. [21] P. Whittle, Adv. Appl. Prob. 12 (1980) 135-153. [22] P. Whittle, Theory Prob. Appl. 26 (1980) 350-361. [23] R. M. Ziff and G. Stell, J. Chem. Phys. 73 (1980) 3492-3499. [24] Stockmayer. [25] J. L. Spouge. [26] J. L. Spouge. [27] J. L. Spouge. [28] J. L. Spouge. [29] J. L. Spouge.

Annals of Discrete Mathematics 28 (1985) 263-304 0Elsevier Science Publishers B. V. (North-Holland)

FLOWS THROUGH COMPLETE GRAPHS W-C. S. SUEN Scliool of hfdienratics, Unioersity of Bristol, Brisrol, England

We consider Ford and Fulkerson network flows [2] through a complete graph G of which the edge czipacitics form a family o f indepcndcnt and identically distributed random variables on [0, co). We study the case when G has vertex sci (0, 1, ..., n - 2 , a ) . ,and edges which are indepeiidently directed so that for each edge joining a pair i, j of vertices,

where r E (0, I]. We obtain asymptotic results concerning the maximum flow whun a typical edge capacity C has distribution p(C>l)=p(lr)(l-F(t)),

It

[ O , co),

where O < p ( n ) < l and F is a known distribution function concentrated on (0,co). The problem studied here is a generalized version of a problem considered by Grimmett and Welsh in [71.

1. Introduction

We begin with a brief description of the concept of flows through a capacitated network. Suppose that G = { V, E ) is a directed graph with vertex set V and edge set E which is a set of some ordered pairs of vertices in V . A capacitated network is obtained from the graph G by associating each edge e = ( i , j ) in E with a nonnegativenumber C,,called thecupaciryof theedge. Letsand t be two special vertices acting as the source and the sink respectively. A feusibfeflow f of value u=u (1) from s to t through G is a non-negative function on E so that the following conditions are satisfied. u if i = s , C f i j - C f j i = - u if i = t , M j , i )E E j : ( i ,j ) E E 0 otherwise,

I

jrj I , we denote by q,,(t) the probability of extinct'on by the m-th generation of the piocess (2,")with a piogenitor of type t in [0, I]. That is,

By conditioning on Z , , it can be shown that

Routine analysis shows that q,(t) converges uniformly to a limit q ( t ) on [0, I ] as rn+co. where q ( t ) is known as the probability of ultimate extinction of the

Flows through complete graphs

269

process {Z,,,).We therefore have that (2.3)

By substituting (2.1) into (2.3), we obtain the following two theorems. Theorem 6. Suppose that r= I . Then q(t)= 1 for all t E [0, I]. Furthermore, if M ( t ) is the size of the entire popiclntion of the process with a progenitor t E [0, 11, Ihen M ( t ) is a geometric random variable with parameter 0, = exp{ -a( 1 - t ) } . That is,

'

P,(M( t ) =j ) = el(I -U, ) j - ,

j = 1 , 2 , ... .

(2.4)

Theorem 7. Stippose that r E (0, 1). We knue two cases: 1

(i) For r # 4, let v = 4 q ( t ) d t . Titen u is equal to the smallest root of the following 0

equation in s: C-urs-

e-crr=e-a

(1 -r)s-

e-a

(1 -r)

,

and q ( t ) is given by

(ii) For r = f , we have

q(t)=q

f o r u l l t e [ O , 11,

where q equals rite smallest root of the following equation in s:

Moreover, we have a critical phetionienon in that there is a constant a, where

(2.5)

W-C. S. Sicerr

270

such ihat, (a) ifcza,, then sup {q(t);tE[O,I]}01,. For t E [0, I ] , let N(t, m) be the s i x o j the m-th generation of’the process {Z,,,} with a progenitor of type t. Then

(2.7)

and if

E

is a constant in (0, l), then

(2.8) Furthermore, fi M(t, m ) is the total number of particles in thejrst m generations of

Flows through complete graphs

27 1

the process, then as m + 03, (2.9)

and (2.10)

Remarks. Readers who would like to explore further into the subjcct of branching processes could consult Harris [3]. It should be noted that Harris considered [ 3 , Cli. 1111multi-type branching processes in a general setting. He placed certain conditions on the process and such conditions, when translated to our process {z,}, cover only the case where r is in (0, I). The rest of the section i s conccrned with flows through branching trees. (See Grimmett and Welsh [7] for flows through Bethe lattices.) The results stated below will only be needed when we prove Theorem 2. For any t E [0, I], let w, be a realization of a branching process with a progenitor t and rate function y given by (2.1). We think of ofas a tree (in which two particles are joined by an edge if they have a parent-child re1ationsh:p) with the progenitor acting as the root of tlie tree. We associate each edge e of L o , with a non-negative random variable U(e) as the edge capacity of c. We assume that tlie random variables { U(e);e is an edge in w,} are independent and have coiiinion distribution function Fconccntrated on (0, m) with finite mean. Let w ’ = o ’ ( q ) be a realization of the edge capacities of the tree (9,.A network f 3 , , t = O m , t ( q , w’) is obtained from the capacitated tree by connecting each ni-th generation vertex of w, to a new vertex co’ by an edge of infinite capacity. We denote by T,,, t = Tm,t(~, r, F ) the niaxinium flow through from the root of w , to the vertex a‘, where the parameters a and r i n Tm,,(cr, r , F ) specify the rate function y (see (2.1)) of the underlying branching process, and the function F is the distribution function of a typical edge capacity. I f w, has become extinct by the m-th generation, then T , , , = O . Suppose that w, and (,I‘ are as defined above. Then we have that for wi2 1,

since a flow in O,n, ,,,must pass through the In-th generation vertices in the tree 0,. Thus thc sequence of random variables, Tm,*= T,,,(u, r, F), converges for every realization w, of the process ( Z m }and every realization w’=w’(w,) of the edge capacities of w,. We let T,=T,(ci, r, F ) be the limit. (Nolc. This gives the definition

272

W-C. S.Sum

of the random variable To(a,r, F) which appeared in Theorem 2.) Writing Z , , , as the set of all particles in the first generation of the process {Z,,,}where Z . = { t } , it is easy to check that for t E [0, 11,

(2.12)

where for X E Z , , , , U, is thecapacityof the edge joining the progenitor t to the is the maximum flow from x to 03' through O m + , , , without particle x, and passing through the edge joining t and x . It is clear that Tm,,is equal to Tm,x in distribution and that when Z,,,=Z#B, the random variables in { F m S xU,X ; x E Z } are independent, Theorem 9. With notation defined as before, we have that

where q ( t ) is the probability of ultimate extinction of the underlying process { Zm) with a progenitor of type t and rate function y given by (2. I). Eg. (2.13) is proved by first noting that

{T,=O}2{z,=@for some m > l } , which implies that P(T,=O)>q(t). This shows (2.13) when aa,,wechoosea sinall positive 6 and delete from w, all edges with capacities not exceeding 6. In so doing, we obtain

where q(6, t ) is the probability of ultimate extinction of a process {ZA} which starts with a progenitor of type t and has a rate function y' satisfy'ng y'(t, x) =(1 -F(h))y(t, x) [or all t , x E [0, 11. Bccause q(6, t ) converges to q ( t ) as Is decreases to 0, Eq. (2.13) is shown. Suppose that o, We next consider :I generalized version of the network and (I)' are defined as before. We for111 a network UAl,, by jo'ning each rn-th generation vertex of w, to a new vertex ix)" with an edge having a capacity U:. We

Flows through complete graphs

273

assume that the family { U i ; x E Zm}is a collection of independent and identically distributed non-negative random variables. We denote by TA,rthe maximum flow from the root of orto the vertex a”through O;,,. Similar to (2.12), we have the following equations.

(2.15)

Theorem 10. With notation dejned as above, suppose that the random variables { U i ) have n common distribution gioen by Y (u: = 6) = 1 - P ( v.; =O)

=p ,

where S E (0, a)and p E (0, I]. Then the sequence ojraiidom variables TL,fconverges in distribution to the mndom variable T,as m-+a3.

The central idea of the proof of Theorem 10 is as follows. For m > 1, let Em,, be the event that the network Oi,,,, has a minimum separating cutset (see [2] for cutset definitions) that contains only edges in the tree 0 J f . For those realizations in which or has an empty m-th generation, we adopt the convention that Em,f has occurred. If Ern,:occurs, then replacing by infinity the capacities of those edges joining the m-th generation vertices to the vertex a”would not alter the maximum flow through O i , f . Therefore, when conditioned on Emsthaving occurred,

Theorem 10 is thus established if lim P ( E m S r ) =1

(2.16)

m-r w

Eq. (2.16) is true because a branching process grows exponentially if it does not die out, and therefore “boLtlenecks” are likely to occur in the early generations.

Remarks. Although we have not been able to find, in general, the distribution of the random variable T,(a, r, F) when a>a,, certain estimates can be obtained. From (2.14), we have that for sufficiently small 6>0,

W-C. S.Suen

274

P(T,>G)=l-q(S,

t)

>mf (1-y(6, t ) ;

=II

t ~ [ oI]> ,

9

which is positive by Theorem 7 for a>cc,. Thus, if the random variables {Vi} defined before Theorem 10 have a common distribution g:ven by

it is easy to show that

Ti,t 1, let A(n, i, h) be the subgraph of G,, induced by all vertices of distance not more than / I from i. We shall obtain certain properties of the graph A(n, i, h) by applying the results stated in the previous section on branching processes. This is done by constructing two rooted trees. We shall be mainly interested in the following two cases:

(i) r ~ ( 0 , l and ) np(n)+ccE(a,, co) as n-+co,

(ii) r = l and np(n)-mE(O, log n

co) as n + m .

Construction C'(m, a(m), r, i, h, q). Given positive integers m,h, and a function q: V,,,x V,,- [0, 11, we construct, for each realization w of the graph G, with parameter6 r and a(m)=mp(m), a tree GT(m, f, h)=GT(w; m, i, h) with root i E V,,,

Flows through conipkte graphs

275

and height not greater than h (where h may depend on m). The tree GT(m, i, / I ) can be regarded as a "random graph" whose randomness is derived from the graph G,,, and from the probability rules of the colouring process set out below in the construction. (i)

The vertex i is coloured red, and acts as the root of the GT(m, i, h).

(ii) We construct each stratum of the tree GT(m,i, h) in turn. Suppose that we have formed the I-rh stratum (I=O, I , ..., 11-1). We form the (/+I)-th stratum by performing a procedure COLOUR' (to be specified below) on each vertex in the I-th stratum, starting fi-om the vertex with the smallest label, and working through to the vertex with the b.ggest label i n :in ascending order. The procedure COLOUR' has an input variablej, wherej is the label of the vertex on which the procedure is performed. Procedure COLOUR'(j). Let S j bc the set of vertices joined to the vertex j by an edge (in G,,,) directed fron1.j. For each k E S j , if k i s not co!o2red, then the vertex k and the edge (j,k ) arc coloured with the red co!our with pro5ability ~ ( kj ),. Each such colouring is do.ie independeiltly of all olher c0lo:lringj. The vertices coloured by the procedure COLOUR'(j) are the offJpring vertices of the vertex j in the tree GT(ni, i, 11). (iii) The colouring process g'ven in (ii) is first applied to the root i to obtain the first stratum, and then to the vertices in the first stratum to form the second stratum. This is continued in the manner set out in (ii) until one of the following occurs: (a) a tree of he'ght h is formed, or (b) the construction gives an empty stratum at some stage.

Construction C"(m, i, h, y,,,). Given a positive integer m, a vertex i~ V,,, and a function y, : [O, I ] x [O, 11-[O, co), let ( Z ~ " ' }be a multi-type branching process i ffl-1 withaprogenitor r,wheret=-ifi#co andt=ifi=oo,andratefunction y,,,.

m ffl For each realization w, of the process { Z ~ " ' }we , shall construct a labelled tree BT(m, i, h)= BT(o,;m, i, h) with root i and he'ght not exceed ng h. To do that, we first associate each vertexje V, with an interval l ( j > ~ [ O I,] so that

[A jil)

f(j)=

-?

~

for j = O , 1 , ..., m - 2 ,

W-C.S. Suen

276

Let J : [0, l]+ V,,, be a function so that J(x)=j

if xEZ(j)

The construction of the tree BT(m, i, h) is as follows. (i) The progenitor is coloured red, and is labelled with i. (ii) We colour the particles in each generation in turn, starting from the first generation. When every particle in a generation is coloured, we proceed to colour the particles in the next generation. To colour the particles in the (l+ 1)-th generation ( I = O , 1, ..., I t - I), we perform a procedure COLOUR" (to be specified below) on each particle in the I-th generation, starting from the particle with the smallest type, and working through to the particle with the b:ggest type in an ascending order. The procedure COLOUR" has an input variable x, where x is the type of the particle on which the pro-edure is performed. Procedure COLOUR"(x). Let S(x) bc the set of all immediate offspring of the particle x. There are two possibilities: (a) If x is red, then we sample each p o k t in S(x), starting from the one with the smallest type. If y E S(x) and the label J(y) has not been used, then the particle y is coloured red and labelled with J(y). If y E S(x) and J b ) has been used, then particle y is coloured blue. (b) If x is blue, then every particle in S(x) is coloured blue. (iii) The colouring and labelling process is first performed to the progenitor so that every particle in the first generation is coloured. The construction is continued by following the method set out in (ti) until one of the following occurs: (a) every particle in the first h generations of the process (Zi'"'} is coloured, or (b) every particle in the process is coloured if the process has died out by the h-th generation. (iv) We think of the red labelled particles as vertices. For each pair of vertices J ( x ) and J(y) (that is, particles x and y), an edge is drawn from J(x) to J ( y ) if and only if the particle x is the immediate parent of the particle y . The resulting graph is named BT(m, i, h). Clearly it is a tree with root i, and has height not exceeding h. The idea behind the construction is as follows. The constructions serve as means

of finding "approximations" to a graph A(m, i, h) by a tree GT(m, i, h) and to a branching process (Zj"'} by a tree BT(m, i, h). The trees GTand BT are random. By choosing suitable parameters in the constructions, we force the trees GT and

Flows through complete graphs

277

BT to have the same distribution. Certain inferences concerning the graph A(m, i, h) can now be drawn by looking at the process {Z;"')]. We now proceed to prove Theorem 5 by applying the constructions given above. We shall make use of three lemmas, prool's of which can be found in the Appendix. Note that by setting h>m, we place no external restriction on the heights of the trees GT(m, i, h) and BT(m, i, 12) because their heights can never exceed m.

Lemma 11. Suppose that r = 1, and that a (m)satisfies

Let 9 : V, x V,,+[O,

I ] be the constantfirnction,

Set h = m and i=O. Let M2(rn,0 ) be the size of the tree GT(m,0, in) obtained from the construction C'(m, a(m), 1, 0, in, 9 ) .Let M , (m, 0) be the total number of vertices joined to the vertex 0 by paths directed from 0 in the graph G,, with parameters r and a(m) as given above. Then

P ( M , ( m , O ) = h f 2 ( n 1 ,o,)= I ,

(3.3)

Lemma 12. Suppose that r = 1, and that a(m) satisfies

Let ym : [0, 11 x [0, 1]+[0, co) be afirnction given by

' 0

orherivise .

Consider the constructioiz C"(nz, i, h, y,). Set h=nz and i = O . Let D(m,0) be the number of particles coloured blue in the construction. Then, as nz-, co, we have

278

W-C. S. Suen

Lemma 13. Let & . ( m y0 ) be the set of all labelled trees of roots 0, and with vertex sets which are subsets of the set V,,,. Consider the constructions C'(m,cr(m), 1,0, myq ) and C"(m, 0, m yy,,,), where a(na), 9 satisfies the hypotheses of Lemma 11 and ym is given by (3.4). Then for each (3 in SZ,(m, 0), we have P ( { W : G T ( ~m; , 0,m)=W})

=Po({o0:B T ( o o ;i n , 0 , m)=W}). That is, both trees have the same distribution. With the help of the lemmas, the proof of Theorem 5 is now straightforward. From this point onwards, we shall sometimes use non-integral quantities in places where integers are required. Such an aberration makes no essential difference to our analysis.

Proof of Theorem 5. We first divide the vertex set V,, of the graph G,, into two sets V,,o and V,,m so that

and V,,, = V,,- V,,,o . Let M I

(respectively M I

) be the number of

vertices in the set V,,, (respectively Vn,), joined to the vertex 0 (respectively co) by paths directed from the vertex 0 (respectively towards the vertex a).It is clear are independent, and that P ( X n = O ) = P ( 3 no path directed from 0 to

00

i n G,,)

of G,, on the vertex set V n , o This . graph cr'(m) 1 where -4has -- vertices with parameters r = l and a' 2 logm 2 as m+co by the hypotheses of Theorem 5. We now apply construction Consider the induced subgraph Gn

2'0

n

d(+, a'(+),

1,0,;.

q ) to the graph G ; , , , where 1 is given by (3.21, and

FIows through complete graphs

219

(q)

to the branching process {Zl } with a progenitor of type 0, and rate function

yfl

as given by (3.4) in which a'(m) is substi-

2

("z )

tuted for a(m). Let M --, 0

1 -i-,

(i

and let D -, 0

C"( +,O,

n

y;).

be the size of the entire population of the process,

be the number of particles coloured blue in the construction Then by Lemmas 11 and 13, we have that

where each inequality holds in distribution only. Since a'(m) -+Iognz

1

2

asm+co,

we have from Lemma 12 that

(;

We note also from Theorem 6 that the variable M -, 0 is distributed as a geometric variable with parameter O(n), where

O(n)=exp

1I [

- --log 1 -

Consider the variable M ,

(5,

m). By relabelling the vertices and by reversing

280

W-C. S. Suen

the edge directions in the graph G,, it will be clear that MI(

c1

tribution similar (identical if n is even) to that of MI -, 0 we have that

i-,

a) has a dis-

.Thus, from Eq. (3.7),

where the random variables Yi and Yi' are independent and have the same distribution as the variable

(- i)

(i ).

M -, 0

Theorem 5 now follows because

and O ( n ) M ( - ! - , 0) converges in distribution to

(l-p(n))-e@"'+exp

an exponential variable with parameter 1.

0

We next show a theorem which will be useful in the proof of Theorem 4.

Theorem 14. Consider the graph G,,, in whiclt r = 1 and a ( m ) satisfies

M (m) __ -9

log 171

a ~ ( 0 00). , Let M I ( m , 0) be the number of vertices joined to the vertex 0 by paths

in G,,, directedfrom 0. Then for each m , M , ( m 0)G Y,,, in distribution, where Y,,,is a geometric variable with parameter B ( m ) given by

O(rn)=

(

I--

*)".:'

Furthermore, if a E ( t ,$), then for any large m ,

E

in

(4, a), we

have that, f o r sl&cienlly

Proof. The first part of the theorem is a direct application of Theorem 6 , Lemmas 11 and 13. (See also the proof of Theorem 5 . )

Flows through complete graphs

28 1

Let M(m,O) be the size of the entire population of the branching process ( 2 ~ ” which ”) starts with a progenitor of type 0 and has rate function ym given by

otherwise.

10

Let D(m, 0) be the number of particles coloured blue in the construction C”(m, 0, m,y,,,). Then from Lemmas 11 and 13,

where each inequality holds in distribution. Since, by Theorem 6, M ( m , 0) is distributed as a geometric random variable with parameter 0(m),where

0 ( m )=(1 -a (m)/m)rn= rn - @ t( o I), Eq. (3.8) now follows from (3.9) and Lemma 12. Note that the error probability in (3.8) can be improved considerably. We next turn our attention to the graph G, i n which r E (0,l) and a(m) satisfies

We collect a result which will be useful in the proof of Theorem 2.

Theorem 15. Suppose that r E (0, l), and that lim ~ ( ( r n ) = a ~ (,a0 0, ) where a, is ni-tm

given by (2.5). Suppose also that h=h(m) is given by

19 log m h = h (m)=.. 32loga-l0gac ~~

Then for any vertex i in the vertex set V,,, of the graph G,, we have the following statements concerning the subgraph A ( m , i, h) of G,

.

(i) If MI (m, i, h) is the number of vertices in the graph A(m, i, 11) then P ( M , ( m , i , h ) > n 1 2 / 3 ) = 0 ( m - 1 / 2 4 ) as m+m.

(3.101,

W-C. S. Suen

282

(ii) If N 1(m, i, h) is the number of vertices of distance h from the vertex i, then for i m-1 t = - if i # m and t=if i= co, we have m m

P ( N , ( m , i, h ) < m 9 " 6 ) = q ( t ) + o ( 1 )

as ni+o3,

(3.11)

where q(x), x E [0, I], is probability of ultimate extinction of the process ( Z , } which starts with a progenitor x and has rate function y, where y is given by

The proof of the theorem makes use of three lemmas. These lemmas help relate the graph A(m, i, h) to the first h generations of a multi-type branching process.

Lemma 16. In addition to the hypotheses of Theorem 15, let q : Vmx Vm+[o, 11 be thefunction given by

r-

. , m

(3.12)

Let A ( m , i, h) be the event that during the formation of the tree G T ( m , i, h) using the construction C'(ni, cc(m),r, i, h, q) there is n vertex j E V,,, such that at least one vertex in S, reinnins uncoloured immediately c@r the procedure CO LOUR'( j ) is performed on the vertex j . Then

Lemma 17. In addition to the hypotheses of Theorem 15, let y,,, : [0, 11 x [0, 13 +[O, a)be a function given by (3.14)

Flows through complete graphs

283

For i E V,, let D(m,i, h) be the number of particles coloured blue in the construction C"(m, i, h, y,) for the tree BT(m, i, h). Then

where t is the type of the progenitor of the process {Zi'")}to which the construction is applied,

Lemma 18. For i E V,,,, let sZ,(m, i, h) be the set of all labelled trees of roots i, with heights not greater than h, and whose vertex sets are subsets of the set V,. Suppose that the hypotheses of Lemmas 16 and 17 hold. Thenfor any W E Q,(m, i, h), we have that P ( { m :G T ( w ; m , i , /I)=&})

=P,({w,: BT(m,; rn, i , h)=G}). That is, both trees have the same distribution. It should be noted that these lemmas are in the same spirit as Lemmas 11, 12 and 13. The proofs of Lemmas 16 and 17 are g'ven in the Appendix. The proof of Lemma 18 is not given as it is similar to the proof of Lemma 13.

Proof of Theorem 15. We first note that if A(m, i, h) has not occurred, then for I= 1, 2, .. ., or h, every vertex of distance I from the vertex i in the graph C, is in the I-th stratum of the tree GT(m,i,h). Therefore, we have from (3.13) that P ( M , ( m , i , 11) >

P ( N ( m , i , 11) < !?a9"

+ 0 (M-~~'), ') = P (N,(m , i , h ) < m9/' ') + 0 ( m 3 / 8 ) , = P ( M , ( m , i , / I ) > rn2I3)

-

(3.16) (3.17)

where N,(m, i, h) and M,(m, i, /z) respectively are the size of the 11-th stratum and the total number of vertices in the tree GT(m, i, h). Let N ( m , i, h) and M ( m , i, h) respectively be the size of the h-th generation and the total number of particles in the first h generations of the process {Z:"'} given i n Lemma 17. Then by Lemma 18, we have that

M , ( m , i , h ) < M ( m , i , h ) i n distribution, and N ( m , i, h ) - D ( m , i, h ) < N , ( m , i , h ) Q N ( m , i , h ) in distributioD.

284

W-C. S. Suen

It follows from Theorein 8 that as m-co

= 0 (m-1 / 2 4 ) ,

(3.18)

and

where q'm'(x), x E [0, I], is the probability of ultimate extinction of the process (2:"')) with a progenitor x and rate function ym given by (3.14). Eq. (3.10) now follows from (3.16) and (3.18). S'nce cc(m)+cc as m-tco, it is easy to deduce from Theorem 7 that q'"'(t)=q(t)+o(l) as nz-co. Thus Eq. (3.1 I ) follows from (3.17) and (3.19). 0

4. Proofs of Theorem 2 and Theorem 4

Proof of Theorem 2. Consider, for any fixed positive integer, I, the subgraphs A(n, 0, I) and A'(n, 00, I ) of the graph G,,, where A(tt, 0, I) is defined as before, and A'(n, 03, I) is the subgraph of G,, induced by all vertices that are joined to the vertex co by paths, directed towards 03, of length not exceeding I. Let M , (u, 0, I ) (respectivelyM i (n, co,I ) ) be the size of the graph A(n, 0, I) (respectively A'(17, co,I)). It is easy to show that as n+co,

This suggests that for any fixed I, the graphs A(n, 0, I ) and A'(n, co,I ) resemble two branching trees for large n. This is the central idea of our proof of the theorem. Suppose that cc(n)=(I+f(n))a, wheref(n)+O as n-+co. Let ( p ( n ) } be a sequence of positive integers tending to co slowly so that the sequences {n-1'24p(n)} and (f(n) p(n)} converge to 0 as n+m. For any labelled and rooted tree w with vertex set V ( o ) , we denote by u(w) the size of the tree. For any vertex i E V,, let QT(n, i, I) be the set of all labelled trees of roots i, with heights not exceeding 1 and satisfy V ( w ) cV,. We write Q',(n, i, l ) as the set

Flows through complete graphs

285

and f ( n , 1) as the set

r(n,I ) =

{(q, wz) :0

1 EQ;ZIy(ll,

0, I ) ,

w2 EQ;.(rl,

03,

I),

and V ( w o ,n ) V(02)=0}. We shall need the following lemmas.

Lemma 19. For (aLE Q,(n, 0 , l ) artd o2E O,(n, co,f), let B ( n , wl, event that A (rz, 0, 1) = o 1 and A' (n , 00 , I ) = ( 02 . Writing

we haue that as n

--f

(02)

be the

co ,

P(B(lt))= 1 -o( I)

Lemma 20. Let y : [O, 11x [Of I]+ [0, co) be a jknction giuen by

Consider the construction C"(n, 0 , I , y ) of the tree BT(n, 0 , I). For w E Q,(n, 0 , I ) , let B ( n , (0)be the event that BT(n, 0 , l)=w and no particle is coloured blue in the cotutruction. Then as I I + cn P o ( B ( n , 0))=1-o(1). 0 E R;(n.

(4.3)

0,I )

Lemma 21. For w E Q,(n, 00, I), let z(w) be the tree obtained from w by relabelling each vertex i in CL) with the label k ( i ) where

and ~ ( O ) = C O , k(w)=O.

Then for (w, ,w 2 )E r ( n , l ) ,

[ y].

where A ( n ) = p ( n ) f(n)+-

286

W-C. S. Siren

Note that (4.1) says that the graphs A ( n , 0, I ) and A ‘ ( n , co,I ) are, with probability 1 -o(l), two disjoint trees, each having a size not more then p ( t z ) . Eq. (4.4) shows that the graphs A ( n , 0 , I ) and A ’ ( n , co,1) “resemb~e”IWO independent branching trees when n is large. Lemmas 19 and 20 are more or less obv ous because the sizes of the graphs A ( H ,O , r ) , A’(n, co,I ) and BT(n, 0, I ) are (almost surely) o f O ( 1 ) as n+m.

Proof of Lemma 21. Let W(w) be the set of vertices of w not i n the I-th stratum of the tree w , and for j~ CY(to), let S j ( ( o ) be the set of offspring vertices of j in (0. Now the event B(n, m,,(02) specifies that (:) there is an edge ( j , k ) for j E W ( w J and k j E s, (wz),

E S,(co,),

or for k E W(uJ and

(ii) there are ~ ( u ( w ~ )1)- (u(wI)-2) pairs of vertices in ojI and ~ ( u ( c o l ) - 1) x ( U ( Q ~ ) - ~ ) pairs of vertices in w2 not jo’ned by an edge in either directions, (iii) there is no edge ( j , k ) f o r j E W ( o , ) and k E V,- V(w,), or for j~ V,- V(w,) - V ( w J and k E W(w2). Let 4 , qn: V, x V,,+[O, 11 be functions g’ven by

tl

r-

n

Then we have that

if i < j ,

Flows through complete graphs

287

giving that

specilies We next turn our attention towards P,(B(n, w,)). The event B ( n , 0,) that forJE V ( Q , ) ,there is exactly one particle of type i n I ( / ) amongst all particles in the first I generations of the process { Z , , } , which starts with a progenitor of type 0 and has rate function y given by (4.2). It is easy to check that

proof of Lemma 21 is thus complete. We now consider flows through trees in a way similar to what we have done in Section 2. For co E RT(n,0, I), we associate each edge E in o with a capacity U ( e ) so that the fam ly { U ( e ) ; e is an edge in o>is a collection of independent random var;ables w’th common distribution F, where P i s g’ven by the distribution of a typical edge capacity of K , . We next form a network Oj(co) by joining each vertex in the I-th stratum of (1) to a new vertex 00’ by an edge of infinite capacity. Let T,(w) be the maximum flow from 0 to co’ through O,(w). We adopt the convention that if o has an empty I-th stratum, then T,(w)=O. For x E [0, a),let

Suppose that B(n,w,,co,) has occurred. It is easy to show that

i n distribution, X,<min(T,(o,), $(o,)) is independent of T,(w,) and has a distribution identical to that where +ft(o,) of 7‘,(r(w2)).Hence, by using Lemmas 19, 20 and 21, we have that

+

P ( X , > x) =P (x,> x ,B ( n ) ) o,(l)

W-C. S. Suen

2x8

whereT,,,=T,,,(u, r , F)isdefinedinSection2and the functions ol(l), ..., o,(l) all tend to 0 as n403. This shows that for x E [0, 03) and for I = 1 , 2 , ... , we have

The proof of part (i) of Theorem 2 is now complete because when uu,, that (4.7)

To show (4.7), we propose to find a lower bound A,’ by constructing a subgraph (2 of K,,. Suppose that the event B(n) has occurred. Then the graphs A ( H ,0, I ) and A‘(n, co,I ) are trees; each having a size not greater than p((n). Let R,(O) and R,(co) be the set of vertices in the I-th strata of A ( n , 0, I ) and A ’ @ , 00, /). Note that if either R,(O)or R,(co) is empty, then X,,=O. Consider the case when neither R,(O) nor R,(co) is empty. Let r,=IR,(O)) and roo=IRl(co)l.Suppose that R,(O)={a(1),

*.., a ( r o ) > ,

and R,(4={a’(l),

... * .’(roo)>

9

where a(l) (1 - E ) p (a - 1) log n )

= P (x, > (1 -E ) p ( a - 1)log n ,Al, A)+ 0 (1)

E~

as small as possible,

Flo HVS througli complete graphs

This implies (4.15), and the proof of Theorem 4 is thus complete.

299

n

Acknowledgements. The author would like to thank Dr. G . Grimmett for his helpful comments.

Appendix A. Proof of 1,emma 11. The Icmnia is obvious because when the function t l ( j . k ) equals 1 for all j , k E V,,,, the construction C’(m, cc(m), 1, 0, i n , t i ) gives a vertex, say k , in the I-th stratum of C T ( i n , 0, in) if and only if the vertex k is of distance I from the vertex 0 in the graph Cn,. 0 B. Proof of Lemma 12. We first note that if Y is a geometric variable with parameter 0, then

Consider the process {Z,(“’}to which the construction C”(in, 0, n i , yo,) is applied. Notice starts with a progenitor of type 0 and has rate function ym given by (3.4). Let that {Z,(”‘)) H be the set of particles in ( 2 ~ ” given ”) by

H = ix : particle s is blue but its parent is not]

For x E H , let n ( x ) be the size of the set containing the particle x and its subsequent descendants in the process {Z:””].Then clearly D ( m , O)= x

Let

1 c,=--+

3

c E

rr(x)

I1

1

~~.We partition the set H i n t o sets

4a

H , , H 2 and I€> so that

H I = {x E H : the particle x has at least one brother whose type is in the interval I ( J ( x ) ) } , H ~ = { X H€ - H I : x < ‘ E ~ } ,

H,= H - HI - Hz

.

For k = l , 2 , or 3, let Dl,(tn)=

1 n ( x ) , giving

that D ( m , O)=D,(ni)+D,(m)+Ds(m).

1e11,

For the rest of the proof, we drop the suffices in the probability measure Po and the expectation operator Eo,writing P and E respectively.

W-C. S. Suen

300

Consider a particle x in the process {Z:""}. Let p x be the probability that the particle gives birth to two or more particles whose types are in an interval I ( j ) for some j E V,. Then by the Markov inequality,

Let B(m) be the event that there exists such a particle. Let M ( m , 0) be the size of the entire population of {Z:""}, and let M ( m , 0, el) be the number of particles in {Z:"'} of types not exceeding cl. Then

Consider particles y 1 and y 2 in the branching process. Let , y ( y l , y z ) be the indicator function of the event that the particles y , and y2 each gives birth to exactly one child of type in f(j) for some j E V,. Then

<m - ' a(my. Since M ( m , 0, 6 , ) (respectively M ( m , 0)) is distributed as a geometric variable with parameter exp(-el a(m)>(respectively exp( -a(m))), we have that

where n, =sup E(n (x) ) . It is easy to show that XEHj

Eows through complete graphs

301

giving that

Therefore, (A.I), (A.21, (A.3) and (A.4) give Eq. (3.5) since D ( m , O)=D,(m)+D,(m)+D,(m).

U

w

C. Proof of Lemma 13. Let j be a vertex in the tree E RT(m,0). Suppose that we are in the process of applying procedure COLOUR'(j) to the vertex j in the construction C ' ( m , a(m), l , O , m , q ) of the tree G T ( m , 0 , m). Let G C ( j ) be the set of vertices coloured immediately before procedure COLOUR'(j) is applied, and let GS(j) be the set of vertices coloured by the procedure. In contrast, consider the procedure COLOLJR"(x) where x is the type of the red particle labelled with f ( x ) = j in the construction C"(rn, 0, m , yn,). Corresponding to the sets CC(j) and C S ( j ) we have sets BC(j) and BS(j) of labels, used before procedure COLOUR"(x) is applied an? used by procedure COLOUR"(x) respectively. Then for k E V,,,- G C ( j ) ,

and for k

E

GCG), we have by construction that

P(kEGS(j))=O.

Also for k

E

V,,, - BC(j),

Po(k E B C ( j ) )= P (particle x gives birth to at least one child of type in I(k))

= 1 -exp

(0

m

otherwise,

by Eq. (3.4). and for k E BC(i), Po(k E BS( j ) ) = 0,

because no label can be used twice in C"(m, 0 . m, ym).

W-C . S . Siren

302

Furthermore, when conditioned on C C ( j ) (respectively EC(j)), the family { { k E G S ( j ) } ; k E V m }(respectively { { k E E S ( j ) ) ;k E is a collection of independent cvcnts. Thus. on the condi*ionthat G C ( j ) = B C ( j ) ,thc sets G S ( j ) and E S ( j ) have thc same distribution. As the trees G T ( m , 0, m) and BT(rn, 0, m) have roots of the sanic label, that is 0, we concludz that Eq. (3.6) holds. 0 D. Proof of Lemma 17. The proof of the lemma is similar t o that of Lemma 12. Consider the process {Z:""} to which the construction C"(m, i, lt, 7,) is applied. Noticc

that {Z:'"'} starts with a progenitor of type t

m

function ym given by (3.14). Suppose that N is a subset of the set of particles in the first generations of the process {Z:"'}, where H is given by

/I

H = ( x : particle s is blue but its parent is not). For X E H , let n ( x ) be the size of the set containing the particlc x and its subsequent dcsccndants in the first h generations of the process {Z:""}. Let 6-5/48 and let h , = J ( l - ~ ) log in x. W e partition the sct H into sets log a - log a, HI = { x E H : the particle x has at least one brother whose type is in the interval I( J(x))), H 2 = { x E H - H I : the particle x is in the first hl generations of the process {Z:"')}}, H , = H - H i - H1.

Let M ( m , i , h , ) (respectively M ( m , i, h)) be the number of particles in the first hl (respectively h) generations of the process {Z:'")}.Then by Theorem 8, E [ M ( m ,i , /

i ) ] = ~ ( r n ~ / ~ ) ) ,

E [ M ( m ,i , h ) * ] = ~ ( r n ~ / ~ ) ,

E [ M ( m , i,

/i1)2]=~(rn1-a).

n(x). By following a method similar to the method

For k = 1, 2 , or 3 , let D x ( m ,i , h)= x e HI

used in proving Lemma 10, we have that, as m-,

03

where n l = sup E ( n ( x ) ) . X 8

Hj

For a particle x in some h,-th generation of the process (2~"'). we have from the basic properties of branching processes that

Flows through complete graphs

303

where M ( m , h - h z ) is the number of particles in the first h - h 2 generations of a process {Z:””) which is the same as the original process {Z;””] except that the process {>:’“’} starts with a progenitor of type x. Thus

n, = sup E ( / Z ( X ) ) I E

Ha

3

Since D(m,i, / I ) =

1 Dk(tn, i, h), Eqs. (A.S), (A.6), (A.7) and (A.8) give Eq. (3.15).

0

I =1

E. Proof of Lemma 16. For any vertex j in the tree G T ( m , i , h), let p ( j ) be the probability that at least one vertex in S, remains uncoloured immediately after the procedure COLOUR’(j) is performed on j. (Note. S, is the set of vertices joined to the vertex j by an edge directed from j in the graph Gn,.) Then by the Markov inequality

Furthermore, if M a ( m ,i , h) is the size of the tree GT(m, i, h), it is easy to deduce from Eq. (2.9) and Lemmas 17 and 18 that

which gives that

References [l] M. Ajtai, J. Komlos and E. Szemerkdi, The longest path in a random graph, Combinatorica 1 (1981) 1-12. [2] L. R. Ford and D. R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, New Jersey, 1962). 131 T. E. Harris, The Theory of Branching Procctses (Springer, Berlin, 1963). [4] G. R. Grimmett and H. Kesten, Random electrical networks on complete graphs, preprint. (1983).

304

W-C. S. Suen

[5] G . R. Grimmett and D. R. Stirzaker, Probability and Random Processes (Clarendon Press, Oxford, 1982). [6] G. R. Grimmett and W-C. Suen, The maximal flow through a directed graph with random capacities, Stochastics 8 (1982) 153-159. [7] G. R. Grimmett and D. J. A. Welsh, Flows in networks with random capacities, Stochastics 7 (1982) 205-229. [8] C. J. H. McDiarmid, General first passage percolation, Adv. App. Rob. IS (1983) 149-161.

Annals of Discrete Mathematics 28 (1985) 305 - 310 0Elsevier Science Publishers B. V. (North-Holland)

ON THE NUMBER OF TREES HAVING A- EDGES IN COMMON WITH A CATERPILL.AR OF MODERATE DEGREES Ioan TOMESCU Furiilty of Mnthenintirs, Uliirersity of Bitcliurest, 70109 Birclirrri~.st.Runialiin

In this paper i t is shown that thc number T(T,; 11, k ) of spanning trees of K. having k edges in common with a fixed caterpillar T, with s edges such that max dcg ( x ) < J E

Y

( r is fixed), satisfies lim T ( T , ;

11,

V

-

(T,)

1 ~ ) / ~ 1 " ~ ~ = 2 ~where / I ~ I.= e ~ ~ lini~ /s/n. / ~ !

n-ro

>I

li

This iniplics that the random variable taking the value k with the probability T(T,; 1 7 , k ) / ~ r " is - ~ distributed asymptotically in accordance with the Poisson law whenever lini sir1 exists. n-r J)

For a connected graph G with at least three vertices, the derived graph G' defined in [ I ] is formed by deleting the endvertices of G. A caterpillar is a tree T whosc derived graph T' is a path. Harary [ I ] proved that for any tree T with at least three vertices, the following statements are equivalent: ( I ) T is a caterpillar. ( 2 ) T 2 is hamiltonian. (3) T does not contain the subdivisioil graph S(K,,,) as a subtree where S(G), or the subdivision graph of G, is the graph obtained from G by inserting just one new vertex on each edge of' G. Consider now a caterpillar T, with s edges e l , ..., e, which is a subgraph of the complete graph K,, with n vertices for n>s+ I . For any selection K of i edges of T,, let K c E ( T , ) = f p , , ..., e,} such that ] K l = i ; these edges span a forest FK of K,, composed of p = ii - i trees. If these trees contain respectively m , , ..., mp vertices (m,+...+ m p = n ) , then it is well known [ 2 ] that the number T ( F K )of spanning trees of K,, that contain FK is given by

n P

T(F',)=

IYijnp-2,

j = I

hence i t depends only on the size of the components of FK and not on their indi305

n rnj P

vidual structure. It is clear that the product

depends only on the choice

j= 1

of K; further this product will be denoted by p ( 9 . Theorem 1. If T,is a caterpillar with s edges such that max deg(x) such that In-hl3r for every a , b E Y, a # b is equal to s-(r-l)(i-1)

(see e.g. [3]), the lower bound follows. Upper bound. Suppose that TTis not the path P, of length s; so there is a vertex x such that deg ( x ) 2 3 . Let u = x y E E(T,)\E(T:) be an edge incident with x such that deg(y)=l. If T s = K , , slet u=xz be any edge incident with x such that u#u. Otherwise, let u=xz E E(T,) n ,?(Ti) and let Wdenote the set of all edges incident with x which arc different froin x y and xz. It follows that WI > 1.

I

On the number of trees having k edges

307

W

u*

.,

O

U

V

0

-9

Y

2

W

FIG. I .

From T, we shall obtain a new caterpillar Uswith s edges by deleting xy and inserting vertex y on the edge xz of T, (see Fig. 1). In Uslet u and ti denote the edges zy and xy. We shall show that

If i= 1 it is obvious that these two sums are equal to 2s since IE(T,)I = IE(U,)l =s. Let i22.In this case (3) holds and the inequality is strict. To see this denote by Sl the left member and by S2 the right member of (3).

It is clear that

c

P(Kl)=

KI:u$KI

c

R2: U $ K z

PW,)

because if we delete edge v = x z from T,, respectively edge u=zy from Us,we obtain isomorphic graphs. Also, by a similar argument we deduce

c

PWJ=

Ki :u,uaKi

K2

c

: u, U E

PWd. K2

It remains to consider the case when U E K ,but u # K I where K l c E ( T s ) ; let K2 denote the set of edges in Uswith the same labels as those in K 1 . Since u=xy $ K2 it follows that vertices x a n d y belong to different components of the forest induccd by K2 in Us;if LY and p+ 1 denote the number of vertices in these two components, respectively, then a 3 I and b2 1 since u=zy E K 2 . It is not difficult to see that ~ ( K i ) = ( a + f i )P d a ( P + 1) P = p ( K 2 ),

I. Tomescu

30 8

where P> 1 denotes the product of the number of vertices in the remaining components, with equality holding only if u= 1. It follows that S,>S, if i22.After some transformations of this type a path P, is obtained, hence

with equality if and only if i= I or T,=P, But the last sum is equal to

C

nt t

... t I I I . ,

ml,

nil =5 + 1

m , ...n z , ,

... ,niq> I

where q = s - i f l . This sum equals the coefficient of x s f l i n the developinent

Theorem 2. For m y fixed k and r let T(T,; n , k ) denote the nirrnber of spanning trees of K,, lzauing k edges in cotmion with a cuterpilfar T, w f f h s edges j b r wliich inax deg(x) r + 2 . Clearly, ( 3 ) follows. For the converse inequality we require firstly the following lemma, which expresses an inequality of FKG type. A proof can be found in ;Ipaper of Harris [6]. See (Graham [4]) for a general view on FKG inequalities.

Avoidance Lemma. Let E be a )nite set and F : 2E-+R a set function satigving the condition Y?Z*F( Y)Z F ( Z ) . Suppose that X is a random subset of' E in which each element is present with probability p , independently of the others. Let A A Z ,..., Ak be given subsets of E. Then we have, for every t ,

Random graphs almost optimalIy colorable

313.

Following current usage, we shall express the above inequality by saying that the conditional distribution of F ( X ) (for the left-hand side conditioning event) is dominated by its unconditional distribution. Returning t o the proof, let us show that we can almost surely find (n/r)(1 -o(l)) disjoint independent r-subsets of vertices in G(n, p,). Clearly this will imply the required inequality X,, O and integer 12 we have the following relation between events:

T,,d h E [ Y, d ( 1 - E ) EY,)V( C 3 E EY,) h

V( v

i=O

( B , ~ ( 1 - 2 & ) ~ ~ - ~ ~ ~ n ~ ~ - 1 ) ) .

W. Fernandez de la Vegu

314

Indeed, it is readily checked that if the left-hand side event occurs, at least one of the right-hand side events has to occur. This implies

+C

P(Bj>(1-2e)EY,-jI

j=O

Tn>j-l). (4)

It is easily checked that, under conditions (1) and (2), we have VarY,=O(EY,) and this implies

p (r,d (1- E ) EY,)= 0 (1).

(5)

We shall prove below that we have

P (C >EEY,) = 0 (1)

(6)

and, for h=(n/r)(1-(4~)”~) and jdh,

I

P (Bj>(l - 2 ~ )EYr-j Tn> j - 1) = o ( l / n )

(7)

This implies, with (4) and (5),

P (T,2 ( n / r )(1 -( 4 ~ ) ’ ~ ~=)1) - o ( I ) , and, as desired,

n in probability, since E is arbitrarily small. Now it remains to prove (6) and (7).

Proof of (6). We have

Because of (2), the bracketed expression is o(1) for l < k < r - 2 . This implies EC= o(EY,) and (6) follows by Markov’s inequality.

Random graphs almost optimally colorable PrOOfOf(7).

315

Let us denote by L j = ( i , , i 2 ,. . ., i,) any admissible set of values for

u &,. i

I , , 12,..., I, and let now Rj denote the set

Let EJ denote the set of

k= 1

edges of the corresponding Kr's and let Fj=[nI2-Ej denote the set of edges of the graph complementary to E,. We claim that the conditional distribution of Bj (relatively to the conditioning event I,= i l , ..., Ij=i,) is dominated by the unconditional distribution of the number B; of' Kr's of H which intersect the set R j . Indeed, the distribution of the set 11, of edges of H belonging to F, coincides with that of a random subset of Fj (with individual probabilities equal to q,,), subject to the condition that, for each i from a collection of indices whose precise definition from L, can easily be written down, it does not contain all the edges of the complete graph with vertex set S,. This implies by the Avoidance Lemma that the distribution of Bj is dominated by the unconditional distribution of the number of Kr's which intersect R, and have their edges i n F, which is in turn dominated by the distribution of the number of K,'s which intersect R, and this concludes the proof of our claim. Let us now consider the additional random variables BJ' and Birrdefined on the same probability space as B j , by

BY= # (K,'s of H which do not intersect R,} and

By'= # {K,'s of H ) . Then BYr= BJ+ B;' while Bi and B;' are positively correlated. Hence Var (B;)d Var (By')= 0 ( E x )

.

We have

forj 1/2 ( p < 1/2) is fixed, then almost all g E G(n, p ) are connected (not connected); if p = 1/2, then P(g is connected)+ l/e as n+co (cf. [3] and [ 131). There are only few rcsults for random graphs in the iz-cube. I n this paper we restrict ourselves to a special covering problem. A covering C of h is a covering of the vertex setofh by subcubes of E" being also subgraphs of h. The number L ( C ) of subcubes in a covering C is called the length of C. A minimal covering of h has minimum length among all coverings of h. This minimum is called the 319

3 20

K . Weber

length of h and is denoted by L(h). A subcube K E h (if h = g , then also the edges of K have to be contained in h) is said to be a maximal subcube of h if there is no K* with K c K * c h . The number of maxiinal subcubes of /z is denoted by S(h). An irredundant covering of h contains only maximal subcubes of / I , and none of them is redundant. I t is clear that L(h)=min L ( C ) taken over all irredundant coverings of h. Deline I!,+ (h)=inax L(c)taken over all irredundant coverings C of lz and denote by T(h) the number of a11 irredundant coverings of h. To construct minimal coverings Tor a g:ven h tlicrc i s a unirying approach: Firs( determine all maxinial subcubes of h, then construct a11 irreclundant coverings or h, and finally find the minimal coverings among the irredundant ones. The complexity of t h i s process may be characterized by parameters :is S(h), L(h) and T(h).If wc renounce the deterniintilion ol'~iiinimalcoverings and replace i t by an arbitrary irredundant one, then the ratio L'.(h)/L(h) yields the possible deviation from the niiniinum length. Obviously, the parameters introduced above are random variables on F(n, p ) or G ( n , p ) , respectively. In this paper bounds for these and similar characteristics are summarized. In the case or F ( n , p ) our investigations were nlotivated by the minimization of Boolean functions. Recall that minimization means to construct disjunctive normal forms (DNF) of ~niniin~im complexity for given Boolean functions. A cusloinary complexity ineasure is the number L ( D ) of conjunctions i n the D N F D (for other complexity nieasures see [14]). Using this ineas~ire,we arrive at our covering problem. In the case of G(n, p ) we note the analogy between the coverings defined above and the well-known coverings of the vertex set of spanning subgraphs of the complete graph by cliques. The last ones are closely related to the chromatic number. Our study was initiated by the work of Glagolev (cf. [4]) and Saposhenko (cf. [6-1 I]). We have used essentially their ideas and constructions.

1. Preliminaries Note that all limits, asymptotics, etc., are considered as n+w. For two sequences a=a(n) and /?=j?(rz) we write a s / ?if a 1/2 there exist opiiinal cl-matchings for d= Llog log 11 - log log log n ] too, but we can notprove itsexistence for d=cl,. Suppojep= 1/2, then d4=[loglogn-logloglogn] and easy calculations show that (3.8) is already not valid for d = d 4 + 2 . Hence Theorem 3.5 g'ves an upper bound exceeding the upper bound from Theorein 3.2(1i)by the factor loglog n; (dl = ~ ~ o g l o g n + l o g l o g l o gfor n ~ p = 1/2). Similar results can be derived for almost all g E G ( n , p ) (cf. [19]). One has to - 112" and replace EV, by EV;, dl by d ; (see Appendix), (3.2) by n-'/'"-'

p N - c p ~ ~ , p N- 2 p p~ *' N - o ( p N ) noncovered vertices a E ~ - ( ( J * u B(f,,f'*))which can be covered by maximal embedded (I-cubes i n at least (EW,( I - o( I))" d ffxent ways. It is clear that, possibly omitting certain members of C , and C2, all these d fferent ways yield different irredundant coverings ol'f. Now let p satisfy (4.1) and put d = d 2 + 1. Then using the bound

(cf. [17]) there is a p*+O such that with cd=d3/EW1+d/EWj,~,2,'+0 and consequently ~ = p N ( 1-o(l)) and f has

irredundant coverings of length a p N ( 1 -o(l)).

Now L + c f ) s p N follows im-

K. Weber

334

IJ I

mediately by the trivial upper bound L + ( f ) < S p N . Furthermore, p*+O can be chosen such that EW,=(EV,)'-"''', and the lower bound in (4.2) is established too. Moreover, we can formulate the following.

Corollary 4.3. ([17]) Siippose the conditions of Theorem 4.1 arc satisfied and put d= (I2+ I . Then t h e c.uist.s n probability p* such thirt f o r alinost all f E F(n, p ) we 11aL.e log T ( f )

N

p N log E W, .

An analogous result can be derived for alinost all g E G ( n , p ) (see [18]). Instead of F(n, p, p * ) the probability space H ( n , p , p * ) = G(n, p ) x E(rz,p*) consisting of all pairs (g,'f'), g E G(n, p), f E F(n, p*), is considered. Thereby the probability of a pair is delined as P r o b ( g , f ) = P ( B ) P * ( ~ )where , P ( P ) is the probability on G ( n , p ) ( F ( n , p * ) ) . EV, and EM', have to be replaced by EV,' and EW;, respectively (see Appendix). A similar role as d, and (I2 play here again the corresponding d i m ns i on s with bar.

Concluding remark. The paper [20] contains ;i discussion o f random graphs in the 12-cube and its relations to other random graph models. Acknowledgements The author wishes to thank the editors for their helpful comments.

Appendix - Notations and abbreviations

Subcube corerings of random graphs in the n-cube

A*=-

II

335

292d

N(1-z)'

Probabilities

1 +x2-' --+o

(+))

Dimensions do = [log n- log I g ( 1 / p ) ] ,

d , = [loglog I I

+ log log log

If

= llog n - log log n -log log( l / p ) 1

- log l o g ( l / p ) l , (1; = [log log I1 - log log( l/p)1

dz = [log log n-loglog( I / P ) ] , ( I ; = [loglog If -log loglog 12 -log log(l/p)l d3

= 1 - loglog( I/P)l, d; = [-loglog( I l p ) - log(- loglog(l/p))l

References [I] M. Ajtai, J. Komlos and E. SzcmcrCdi, Largest random component of a k-cube, Combinatoricn 2 (1982) I , 1-8. [2] P. Erdos and A. R h y i , On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci., Ser. A, 5 (1960) 17-61. (31 P. Erdos and J. Spencer, Evolution of the n-cube, Coniput. Math. Appl. 5 (1979) 33-39. [4] W. W. Glagolcv, Some estimations for disjunctive normal forms of Boolean functions, Problemy Kibernet. 19 (1967) 75-94 (in Russian). [ 5 ] A. D. Korshunov, On the length of the shortest disjunctive normal form for Boolean functions, Diskretny Analiz 37 (1981) 9-41 (in Russian).

335

K. Weber

161 A. A. Saposhenko, Disjunctive Normal Forms-Metric Theory (Moscow, 1975) (in Russian). [7] A. A. Saposhenko, On the greatest length of an irrednndant disjunctive normal form of almost all Boolean functions, Mat. Zamctki 4 (1967) 6, 649-658 (in Russian). [8] A. A. Saposhcnko, On the complexity of disjunctive normal forins obtaining by a greedy algorithm, Diski-etny Analiz 21 (1972) 62-71 (in Russian). [9] A. A. Saposhenko, Lectures about niinirnizalion theory (Moscow, 1973) (unpublished). [lo] A. A. Saposhenko, Metric properties of almost all Boolean functions, Diskretny Analiz 10 (1967) 91-1 19 (in Russian). [I I ] A. A. Saposhcnko, Geometric striic(ure of almost all Boolean functions, Problcmy Kibcrnet. 30 (1975) 227-261 (in Russian). [I21 E. Toman, Geometric slructure of random Boolean functions, Problemy Kibernct. 35 (1979) 111-132 (in Russian). [I31 E. Toman, On the probability of connectedness of random subgraphs of the n-cube, Math. Slovacn 30 (1980) 3, 251-265 (in I<ussian). [14] K. Webcr, On some conccpts for niinimality of disjunctive normal forms, Problcmy Kibernet. 36 (1979) 129-158 (in Russian). [ I 51 K. Weber, T h e length of random Roolean functions, Elektron. 1nform;itionsverarb. u. Kybernet. EIK 18 (1982) 12, 659-668. [16] K. Weber, Subcubes of random Boolcan functions, EIK 19 (1983) 7/8, 365-374. [I71 K. Weber, Prime implicants of random Boolean functions, EIK 19 (1983) 9, 449-458. [18] K. Weber, Irredundant disjunctive normal forms of random Boolcan functions, EIK 19 (1983) l O / l l , 529-534. [19] K. Webcr, Subcube coverings of random spanning subgraphs of the n-cube, Math. Nachrichten 120 (1985) 327-345. [20] K. Weber, Random graphs - a survey, Rostock. Math. Kolloq. 21 (1982) 83-98. I211 K. Weber and B. Klipps, Boolean functions of msxirnum length and Sperner-type conditions about the set of faces of the n-cube, EIK 19 (1983) 3, 187-193.

Annals of Discrete Mathematics 28 (1985) 337-348 0 Elsevier Science Publishers B. V. (North-Holland)

RANDOM GRAPHS AND POLYMERISATION PROCESSES Peter W HTTTLE Sfutistiml Laboratory, Utiiiwsity of Cnnibridge, Cumbridge, Eirylund

It i s shown that many distributional problems for randoni graphs (and so for polynicrisation processes) can bc quickly resolwd by appeal to a cliissic combinatorial identity. I t is also sIio\\m that certain nntural eqtrilibriuni distributions rot- configtlration in a n arc-building'breaking process also supply the time-dcpendent solution for a stochastic model of pure arc-building.

1. Introduction A graph on N labelled nodes can be regarded as a pattern of bonding between N distinguishable particles, or units. It thus provides a representation of a pattern of interaction between these units in which bonding or its absence is the only relationship, notions such as physical distance or interaction as a function of distance being absent. This skeletal version of interaction is that generally adopted in the literature on polymerisation; the study of the molecules formed by bond-forination between units. The nodes of the graph represent units, and are supposed distinguishable, i.e. labelled. The nodes of the graph may be cofoured if different types of unit are possible. The arcs of the graph represent bonds. If the arcs are directed then they indicate a directional property ol' the bond; perhaps that of being itiiiiuted by the unit from which the arc emerges. Multiple bonding and self-bonding are not excluded, in general. The connected components of the graph represent molecules, or poljnicrs. Suppose that one allows the bonding pattern to be random, so that one is effectively considering random graphs; i.e. a probability distribution over the set of possible graphs on the N assigned nodes. Then there is considerable interest in the nature of the statistics of the polymers induced by the prescribed statistics on bonding. This interest has been followed in both the mathematical and the physical literature, i.e. in the literature on random graphs and in the literature on polymerisation. 337

338

P. Whlffle

The prescription adopted most often in the random graph literature (see e.g. Erdos and RCnyi [I], Erdijs and Spencer [2], Stepanov [3, 41) is that only single, undirected arcs between distinct nodes may occur, and occur independently (for distinct node-pairs) with constant probability p. This may seem like the simplest prescription, but is not (see Section 3 ) . On the other hand, it is much simpler than physical realism would demand (see Section 4). However, it does produce the important phenomenon of a phase traiisition. If we set p = c / N then for N large there is a critical value C of c. For cC the probability approaches unity that the most ofthc N nodes are conncctecl; i.e. belong to a single polym-r molcculc, whose sizc must then be of order N . This pheiiomcnon is the sollgcd transition or gekition point observed by polymer scientists, and explained by various workers. The model used for the cxplanation has varied, but the mathematics cmxgingseern to be remarkably consistent. Flory [5], Stockinayzr [6,7] and Goldberg [8,91 assum:d that a parameter mc:isuring degree of bonding increased dcterminislically with time, and that the statistics of the process adapted on a shorter time-scale to the equilibrium distrihution predicted by statistical m-\chanics. Watson [lo], Good [ I I ] and Cordon [12, and many later publications, e.g. 12a] saw the process as a spatial branching process. Whittle [13, 141 took ;i reversible Markov model of association and dissociation ( i s . bond formation and breaking) and exmiined its statistical equilibrium. All these models demonstrated g-lation, and evolved similar mathematics despite their diffxing starting points. However, they were also all restricted to polymzrs which were trees (i.c. ring formation was excluded). Whittfe [15-171 showed how this restriction could be removed, with an actual simplification of treatm"nt. However, the radical simplification came in a subsequent papcr (Whittle [18]) in which it was shown that the equilibrium polymer statistics could be deduced from plausible equilibrium statistics for a complete description of the graph by appeal to a classical conibinatorial lemma.

2. The fundamental identity and its application If there is no question of units changing their type (i.e. of nodes having a variable colour) then the configuration %' of the graph is described coinpletely by the pattern of directed arcs between the N labelled nodes. That is, by s = ( s a b ) where sob is the number of arcs f*rom node a to node b (a, b= I , 2, ..., N ) . BY we indicate a summation over all configurations for prescribed N , by we 9

1 N

indicate a summation over N = 1,2,3, ... . The lemma is one that appears in various contexts in combinatorics, and for the case o1'tl.e mumeration of graphs can be given the following expression.

Random graphs and polymerisation processes

339

Lemma 1. Suppose that to each graph % on an arbitrary number of labelled nodes can be attached a weight w(V) with the properties: (i) that w(U) is invariant under permutation of the nodes, and ( ' i ) that w(Ce)= w('e,)w(V2)if%can be decomposed into mutually unconnected graphs V, and g 2 .Tlien

where the sum C' covers all distinct connectedgrphs, N is the nuniber of nodes of @, and the tcwn f o r N = 0 is tuken us unitjj.

For proofs of the fundamental identity (1) we can reler to, for example, Percua [19, p. 521. Suppose now that PN(.s)is the probability of configuration s on the given N nodes, and that

where i indexes the possible types ofconnected graph (i.e. of polymer at the completest level of prescr:ption) and mi is the number of times component i occurs in the graph. The proportionality constant in ( 2 ) will be a function or N alone which normalises the distribution over s for fixed N . The weighting Q,v(s) evidently satisfies the conditions of the lemma, and identity ( I ) then becomes

Now, typically, one will classify polymers at some lower level of description, indexed by r, say. We shall g've examples in Szctions 3 and 4. We shall refer to a polymer of type r as an r-mer. Presumably the description will also determine the number of units in the r-mer;let us denote this by R . Let r ( i ) and X ( i ) denote the polymer type and size of a component i and let us define gr=

C

Vi.

i:

r(i)=r

Let us denote the numbcr of r-mers in the configuration by n,, and define the probability generating function Z7,(z) = E

(n

z:r) .

r

This defines the polymer statistics, which we assume to be the object of interest.

P. Whittle

340

Theorem 1. Assume the distribution of complete configurations given by (2). Then n N ( z ) is proportional to the coeficient of uN in tlze expansion of

in powers of ihe scalar 0. Proof. We take the identity (1) with

and obtain

The coefficient or ON in the left-hand-member is indeed proportional to n N ( z ) .

0

It is not necessary that either of the infinite series in (3) should converge for any 0. It is simply asserted that coeficients of ON on the two sides are cqual for all N . However, if the series do converge for some 0, then the identity has an interesting interpretation. It asserts that there is an opcn version or the process (i.e. a version in wh'ch Nis also a random variable) in which then, are independent Poisson variables. The two conditions of Lemma 1, which lead to this conclusion, are: (i) the units are statistically identical, even if distinguishable, and ( i i ) the appearances of polymcrs would be independent events were it not for constraints on N . The solution for n,(z) yielded by Theorem 1 is moderately explicit if the cocfficients g, can in fact be calculated. What can indeed often be calculated in the sun1

G (0) = ORg,=C OR"'y/, .

1 r

i

If we set all the z, equal to unity i n identity (3) then this reduces to

where

Random graphs and polymerisation processes

341

and theseexpressions are often computable. It may then be that g, can be calculated from C(0) if the yi’s include variables which can serve as marker variables. Indeed, the appropriate level of description of the polymers is fixed by the level at which such marker variables exist, and g, is extractable from G(0). Otherwise expressed, one distinguishes just those polymers which are energetically distinguishable. (Use of the term “energy” is a reference to statistical mechanics. A factor in the configuration distribikon which reflects interaction can be regarded as relaled to the potential energy of the configuration.) We shall give examples in the next two sections.

3. Poisson bonding Suppose that we choose

Here V is the “volume” of the region within which the N units are constrained, and we assume that

where 1is a parameter measuring intensity of bonding. The factor P” included i n (5) is constant in that it is independent of s, but it becomes important when we consider a distribution of polymers over more than one “compartment” in space. The assumption impl,cit in (5) is that the N 2 quantities subare independent Poisson variables, each with expectation S/2. Later we shall consider the thermodynamic limit, in which N and V become infinite, the density p = N / V being held fixed. The point of the dependence (6) is then that the total expected number of bonds issuing from a prescr,bed unit remains fixed at Nh=Ap. This Poisson assumption seems the simplest possible: polymer statistics are’ purely combinatorial in that all polymers with R units and L bonds have the same energy.” These two statistics then constitute the natural description level for polymers in this model. The Poisson assumption leads to more natural mathematics than the assumption of independently occurring single undirected bonds mentioned in the Introduction. The assumption allows the phenomena of selfbonding and mult:ple bond‘ng, which may seem unnatural. It is not clear that they are. but their occurrence has negl gible effect in the thermodynamic limit. ‘1

342

P. Whittle

We have

so that

The series (8) defines G(0) as a power series in OV and 1/V,

where roRLis a purely combinatorial term. The pair R, L constitutes the natura prescription r at this level, with 0 and E, being the marker variables for R and L respectively. We have the identification

Note now that

with equality if and only if the polymer is a tree. We see then from (10) that tree polymers have an abundance of order V while other polymers (let us call them ring-polymers) have an abundance of lower order, the order decreasing with the amount ofcyclisation ofthe polymer. One can say that there are so many ringpolymers that the abundance of any given one must be low, although their total effect may be appreciable. Note that the series (8) always diverges, as must then the series (9). This is because of the multitude of contributions from ring-polymers. If one considers the contributions t o these series from tree-polymers alone then the series converge so long as 8 does not exceed a value which is related to the phenomenon of gelation. Asindicatedin [I61 by far theeasiest way to detectgelationis toexamine the quantity PN+nPN-nas a function of n where PN= &IN!. This quantityis the probability that 2N units should distribute themselves over two disjoint reg’ons each of volume V as ( N + n , N - n ) , bonding being allowed only within each volume

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343

according to the prescription (5). In the sol-state PN+,,PN-,, has its maximum at n=O (corresponding to statistical equi-distribution between the two regions): in the gel-state it has symmetric maxima at n= +no say (corresponding to a tendency for the units to clump in one region or the other). Using then the criterion

as a criterion for gelation and the evaluation (7), we find that the condition for gelation in the thermodynamic limit is

This agrees with the value deduced by much less direct arguments.

4. Bond interactions

Let a unit which has formedj bonds (in either direction) be denoted an a, and let n, be the number of 0,’s in the N given units (j=O, 1, 2, ...). Suppose the Poisson specification (5) now modified to

The assumption is then that the basic Poisson statistics are modified by bond-number dependent terms, the term H, being h:gh or low according as a unit w i t h j bonds is relatively favoured or disfavoured. Alternatively expressed, the energy of the configuration is a function only of the bond-numbersj, and is low or high according as H, is high or low. It is shown in Whittle [IS] that QN now has the evaluation

where

The natural description r of a polymer is now r=(ro, rl, r z . ...), where r, is the number of aj’s the r-mer contains. The H , are natural marker variables for the T i .

P. Whittle

344

So still are 0 and 6 for R and L, although these are unnecessary in that the r, determine R and L

The series (4) with the evaluation (14) for QN determines G (0) as a power series in UV, A/V and the HJ G(O)=

CarV R e L I L O R (HI') n , r

i

where or is purely combinatorial, and wc have

We use the criterion (11) for criticality, with Q N having evaluation (14). In the thermodynamic limit this yields the criterion

for g:lation, where

5 is determined by

All this work can be generalised naturally to the case of several unit-types, and to the case when there is a d fferential effect between inter- and intra-polymer bond ng - an effect which encouragzs or d.scourages cyclisation (see Whittle [ I 7, IS]). Reference might be made to the paper by Spouge [20] which relates the methods of this paper to Gordon's branch ng process model. It should perhaps be emphasised that in the configuration s of (5) and (13) nodes are distinguishable, the ends of a loop are distinguishable (since the loop is directed), but the sob bonds between g'ven nodes a and b are not distinguishable. It is just because of this latter fact that the (sob!)-' term occurs in ( 5 ) and (13).

Random grnphs and polymerisation proresses

345

5. Equilibrium solutions as time-dependent solutions

Flory's original approach was not to consider the equilibrium version of association and dissociation, but rather to consider the time-dependent vers:on of a purely associative process. That is, a process i n which bonds form b u t d o not break. He then deduced the occurrence of gelation at a time when the degree of bonding had reached the critical value. This view expresses the experhenceof the polymer chemist: when physical conditions are created which favour polymerisation, then polymerisation proceeds progressively, and, after some time, gelation is observed as a rather definite event. Flory's inathematical approach was a mixture of statistical mechanics and heuristics. However, one is impelled to ask whether equilibrium distributions such as (5) and ( I 3) might not be seen as time-dependent distributions for an appropriate stochastic process of pure band.ng. Consider a time-dependent version of (5)

Here the bonding parameter S=A/V has been replaced by k / V . We have also dropped the N subscript for simplkity. Distribution (15) implies that the Sub are independent Poisson variables with expectation it/2Vat time t. This is exactly the distribution that would hold for a process, completely dissociated at t = O , in which for any a, b there is a fixed probability intensity/1/2Vthata new bond will form from unit a to unit b, independent of the previous history of the process. This is the simplest stochastic bonding process one can ilnag.ne, and one sees from (12) that it gels at time

Let us write the transition in which sob is increased by one as s+s+euh, and denote the correspond ng transition intensity as A (s, sscub). Then the Poisson model we have just considered is characterised by

/i(s,s+eu,)=A/2V

(a, b=1,2,

these being the only transitions possible.

... , N )

P. Whittie

346

Consider now the more general distribution

in which the time-dependent Poisson statistics are modified by a time-independent term @(s), representing interaction effects associated with the potential energy of configuration s. This can indeed be seen as the time-dependent solution, either exactly or in the thermodynamic limit, of a natural stochastic bonding process.

Theorem 2. Consider the pure bonding process for which

and let P(s, t ) denote the distribution of s at time t for rhis process, given complete dissociation at time t = 0. Let (18)

denote the normalised version of expression (16). Then P, P* respectively satisfy

ap*(s,t )

--

at

-XI P*(S - cab, t ) A (sa

cab, s) -A*(t)P*(s,

t),

b

where

I

A*(t):= P*(S , t )A ( t ) . S

If for eoery t

A (s) is constant (in s) for all s such that P*(s, t ) > O then P(s, t)=

P*(s, t ) .

Proof. Equations (19), (21) constitute the usual Kolmogorov forward equation for the Markov process specified by (17). One readily verifies that expression (16)

Random graphs and polymerisation processes

347

satisfies

and hence that its normalised version satisfies (20). The two equations (19), (20) evidently agree under the condition stated (and only under an almost-everywhere version of it). More explicitly, suppose we have established that P ( s , T) = P*(s, T) for z < t . To establish equality of rate of change we then require, by (19) and (20), that

( A (s) -n*(t))P*(S , t ) = 0 which is equivalent to the condition of the theorem.

0

The condition is quite a restrictive one. Unless there are configurations which are actually forbidden then P*(s, t ) > O for all s and all t>O, and the condition of the theorem would require that

be independent of s. However, one can imag:ne that, under appropriate conditions on @, the random variable a(s) would converge (in the appropriate probabilistic sense) to a constant in the thermodynam'c limit under the distribution P*(s, t ) , for fixed t. That is, that the condition of the theorem is satisfied, and so P=P*. in the thermodynamic limit. Justification of this assertion obviously requires a careful argument, which I hope to supply in a later publication. Van Dsng:n and Ernst [21] consider a determ'nistic polymerisation model with reversible rates of a particular form; and show that for this model some non-equilibrium solutions are indeed equilibrium solutions conditioned by a kinetic parameter.

References [ l ] Erdos, P. and RCnyi, A., On the evolution of random graphs, Mat. Kutat6. Int. Kozl. 5 (1960) 17-60.

[2] Erdos, P. and Spencer, J., Probabilistic Methods in Combinatorics (Academic Press, 1974). [3] Stepanov, V. E., On the probability of connectedness of a random graph, Teoriya Veroyatnostei i ee Prim. 15 (1970) 55-67.

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P. Whittle

[4] Stepanov, V. E., Phase transitions in random graphs, Teoriya Veroyatnostci i ee Prim. 15 (1970) 187-203. [ 5 ] Flory, P. J., Principles of Polymer Chemistry (Cornell University Press, 1953). [6] Stockmayzr, W. H., Theory of molecular size distribution and gel formation in branched chain polymers , J . Chem. Phys. 1 1 (1943) 45-55. [7] Stockmayer, W. H., Theory of molecular size distributions and gcl formation in branched polymers. 11. Gcneral cross-linking, J. Chem. Phys. 12 (1944) 125-131. [8] Goldberg, R. J. J., A theory of antigen-antibody rcactions. I., J. Amer. CIicm. Soc. 74 (1952) 5715. [9] Goldberg, R. J. J., A theory of antigen-antibody reactions. II., J. Amer. Chem. SOC. 75 (1953) 3127. [lo] Watson, G . S., On Goldberg’s theory of the precipitin reaction, J. Immunology 80 (1958) 182-185. 1111 Good, 1. J., Cascade theory and thc molecular weight averagcs of the sol fraction, I’roc. Roy. SOC.A 272 ( I 963) 54-59. [I21 Gordon, M., Good’s theory of cascade processes applied to the statistics of polymer distributions, Proc. Roy. SOC.A 268 (1962) 240-259. [12a] Gordon, M. and Temple, W. B., Chemical Applications of Graph Theory, ed. A. T. Balaban (Academic Press, 1976) 300-332. [13] Whittle, P., Statistical processes of aggregation and polymerisation, Proc. Camb. Phil. SOC.61 (1965) 475-495. [I41 Whittle, P., The equilibrium statistics of a clustering process in the uncondensed phase, Proc. Roy. Soc. Lond. A 285 (1965) 501-519. [15] Whittle, P., Polymerisation processes with intra-polymer bonding. I. One type of unit, Adv. Appl. Prob. 12 (1980) 94-115. 1161 Whittle, P., Polymerisation processes with intra-polymer bonding. 11. Stratified processes, Adv. Appl. Prob. 12 (1980) 116-134. [I71 Whittle, P., Polymerisation processes with intra-polymer bonding. 111. Several types of unit, Adv. Appl. Prob. 12 (1980) 135-153. [I81 Whittle, P., A direct derivation of the equilibrium distribution for a polymerisation process, Teoriya Veroyatnostei 26 (1981) 350-361. (191 Percus, J. K., Conibinatorial Methods (Springer, 1971). [20] Spouge, .I.L., Polymers and random graphs: asymptotic equivalence to branching processes, J. Stat. Phys. (1984) (to appear). [21] Van Dongen, P. and Ernst, M. H., Pre- and post-gel size distributions in (ir)rcversible polymerisation, J. Phys. A: Math. Gen. 16 (1983) L327-L332.

Annals of Discrete Mathematics 28 (1985) 339-359 0 Elsevier Science Publishers B. V. (North-Holland)

CRITICAL PERCOLATION PROBABILITIES * John C. WIERMAN

This papcr sumninrizes tlie current state of Itnowledge of' critical probabilities in percolation niodcls. Tlicre are four coiiimon definitions o f critical probability in the literature, which are known to be unequal for soinc graphs. A heuristic method of Sykes and E s i m [I71 has produccd corrcct critical probability values for a fcw planar lattice models. klowcver, counterexamples have been coixtructcr! to some conclusions of the Sykes and Essam niethod, and (hc niethod is valid only for two-dimensional graphs. Exact critical probabilities have been rigorously determined for a few two-dimensional percolation models, using techniques of Seymour and Welsh [I 51, Russo [14], and Kzsteii [lo] involving crossing prob.ibilities of rcctangles.

1. Introduction

Percolation theory developed from B probabilistic model for fluid flow in a medium proposed by Broadbent and Hainmersley [I]. Their niodcl associated the random mechanism only with the ~nedium,with no randomness associated with the fluid as in the more familiar dlffusion models. Consequently, percolation models typically exhibit drastic changes in behavior as a parameter threshold is crossed, which leads t o their popularity a s models for the study of phase transitions and critical phenomena in statistical mechanics. There has bcen dramatic growth in the physics literature on percolation in recent years, consisting primarily of conjectures, numerical evidence, and a wide range of specific applications. Substantial mathematical progress has bcen made, providing rigorous deterrnination of the critical probability in several models. New tools, such as the FortuinKasteleyn-Ciinibre correlation inequality and sub-additive processes, originated within percolation theory. However, most results apply only to two-dimensional models, and there is little knowledge 01' models in three or more dimensions. The medium is represented by a graph 9' in a percolation model, from which sites or bonds are deleted at random to form a network of channels through which

* Research supported by the National Science Foundation under Grant No. MCS-83032 38. 349

3 50

J. C. Wirrniun

fluid may flow. Early research centered on two standard models - the bond percolation model and the site percolation model. In a bond percolation model on a graph 9, each edge of 9 is randomly open with probability p , o , < p d l , the parameter of the model, and closed with probability 1 - p , independently of all other edges. Fluid may pass through only open bonds. I n a site percolation model on 3,each vertex of 9 is randomly open with probability p , O < p b I , and closed with probability 1 -p, independently of all other vertices. A bond is open (closed) if and only if both its endpoints are open (closed) sites. Thus, i n a site model, some bonds, those with one open and one closed endpoint, are neither open or closed. For either niodcl, let P, and E, denote the corresponding probability measure and expectation operator on configtirations of 9.In either sile or bond models, the focus is on the extent of fluid flow possible. Of particular interest, if 9 is infinitc, is the probability that the fluid reaches an infinite set of sites from a single fluid source site. Fisher [5] introduced a bond-to-site transformation which converts a bond percolat~onmodel on