Table Of ContentRANDOM 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
Admi Mickiewicz University
Poznari, Poland
1985
NOR.rH~HOLLAND- AMSlERDAM . NEW YORK . OXFORD
0 Elsevier Science Publishers B. V., 1985
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Library of Congress Catalogiu~i n 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 Uni-
versity, Pornad, Poland”
A selected collection of papcrs based on lectures presented at the 1st
Poznari Seminar on Random Graphs held at 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 0-444-87821 I
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 in 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 in Poland in 19S3. The first seminar, to
which a representative group of researchers was invited, took place in 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 contrib-
uted to a fruitful exchange or opinioiisaswell as to the initiation of many pro-
fessional 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 Mickie-
wicz 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
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. Szaiiikolow icz, 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
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
in 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 o f u
vertices or an independcnt set of u vertices. The well known proof ol Szekeres
gives
( ::y ' ) (;I;).
r(u, v)< and in particular r(n , n)q
No non-trivial lower bounds were known. In 1946 (after many failures) it occurred
to me to try non-constructive methods it. consider all possible graphs on m
labelled vertices (there are clearly )'(2 such graphs), a very simple computation
shows that if
n
m < cn22
then the number of graphs on m vertices which contain a K(n) (i.e. a complete
(''"1
graph of n vertices) or an independent set of n vertices is o(2 ). This of course
implies
1
2 P. Erdos
By using a lemma of Loviisz, Spencer improved the values of the constant in
(2), but as far as I know
I (It , 11)
lim-,--=co
n22
is still open. I offered (and offer) 100 dollars for the proof of the existence of
1
lim r(n, n>” and 500 dollars for the determination of the value of this limit (this
latter offer probably violates the minimum wage act). The limit if it exists (it
surely docs) is between 2”’ and 4.
My next success with the probability method was to give a good lower bound
for 43,n ). 1 was always sure that r(3, n)=o(n2)a nd _r(_3’ _n) -+co but for many
n
years 1 could get nowhere. In November 1956 in the bus travelling from the Dow-
kers (Yael and Hugh) to University College London I realized that by a construc-
tion in n-dimensional geometry I can show that r(3, Ii)>nl+‘. By a fairly corn-
plicated application of the probability method (later simplified by Spencer)
I showed that
A few years later (1966) Graver and Yackel showed
cn
r(3, n)< -1oglogn.
log n
I offered 50 dollars for r(3, n)=o(n2). They also used the probability method.
Recently Ajtai, Komlds and Szemer6di proved by a new method (which has many
other applications and is also probabilistic in nature) that
cn2
r(3,n)c--.
log n
An asymptotic formula for r(3, n) is nowhere in sight at the moment.
I was (and am) sure that r(4, n)>n3-& and more generally r(k, n)>nk-e for
every fixed k, and was very dlsappointed that the probability method gives only
a much weaker result. Several other mathematicians tried without success this
natural and attractive conjecture. 1 offer 250 dollars for a proof or disproof
of r(4, ~l)>n~t-h~e ,d ifficulties are perhaps not only technical in nature.
Welcoming address 3
Very few Ramsey numbers are known, r(4,4)= IS was proved by Greenwood
and Gleason in 1955. r(3, n) is known for all n Q9 except n = 8. I sometimes make
the following joke in my lectures: Suppose an evil spirit would tell us, “Unless
you tell me the value of r(5, 5) I will exterminate the human race.” Our best
strategy would perhaps be to get all the computers and computer scientists
to work on it.If he would ask for r(G,6) our be,t bet would perhaps be to try to
destroy him before he destroys us (unfortunately we are very good at destroying
[especially ourselves]). If we could be so smart as to be able to compute these
numbers without computcrs, we would not have to pay any attention to the evil
spirit and could tell him “just try and see what will happcn to you.”
For the sake of historical accuracy I have to add that in fact several years
before the work on r(3, n) I proved by probabilistic methods for every k and I
there is a graph of chromatic number k and girth 1. Lovrisz and later NeSetEil
and Rod1 obtained a constructive proof of this resull. As far as 1 know there is
no constructive proof for the following result of mine (which follows quite easily
from the probabilistic method): For evcry k there is an &k so that for every n>
n,(c, k) there is a Ci(n) i.e. a graph of n vertices of girth > k for which the largest
independent (or stable) set is </z’-(a~n d therefore of course its chromatic number
is >ne).
Let me now tell you two “spectacular” (?)s uccesses which I had with the prob-
ability method. In 1962 Professor Schlitte at a meeting in Oberwolfach asked
me the following question: Is it true that for every k there is a tournament for
which every set of k players is beaten by one of the other players? Denote by
nk (if it exists) the smallest number of players in such a tournament. Schutte
observed that nl=3, n2=7, but he d’d not know if rz3 exists. In the language
of graph theory a tournament is a directed complete graph. Schutte’s problem
asks for a directed complete graph in which for every set of k vertices xl, ... , xk
there is a y so that all the edges are directed from y to xi, i= 1,2, -..,k .
At first 1 was baffled by this problem but then while I was resting for a few
minutes after lunch it occurred to me to try the probability method and to direct
the edges at random. Sure enough I proved without much trouble that for every
n>ck2zk there is such a tournament on n players. In other words nk<ck22‘.
n, >2k+1- 1 is easy and a few years later Esther and George Szekeres proved
nk>ck2k and they also showed n3= 19. An asymptotic formula for nk is nowhere
in s;ght at the moment and the value or n4 is also unknown.
The following interesting problem remains open. Is it true that for every n>nk
there is a tournament on n players in which every set of k players is beaten by
another player. I was never able to get anywhere with this simple and attractive
problem. Pcrhaps I overlook a simple argument but perhaps this problem can
not be attacked by the probability method.
My second triumph occurred at dinner at St John’s College in the summer
4 P. Erdiis
of 1971. Professor Mordell who invited ine told me, “I know you are willing
and eager to talk Mathematics at any time, sit down next to my young colleague,
he is working in functional analysis and he has a coinbinatorial problem whose
positive solution would be useful for his work.” The problem stated as follows:
Let there be g ven an n by n matrix all whose entries are 0 or 1. Assume that the
number of 0’s is >cn2 where O<c<l is a constant independent of ti. Is it then
true that there is a rectangle of u coluinns and u rows which consists mtircly of
U’V
0-s and for which ---+aH.e re I could give the answer berore I finished my
n
soup. The answer is negative and in fact for almost all such (0, 1) matrices
max u * u=( 1 +o (1))max rc‘n. (7)
r
(7) follows quite easily by the probability method. The unhappy ending is
that the negative answer was of no usc for the problcm of my colleague.
As a triumph I could also mention our disproof with Fiijtlowicz (in this case
he suggested the probabilistic approach) of a well known conjecture of Ha$.
Here Callin earlier disproved the conjccture by a simple exiiinpk, the point
of our disproof was that we disproved the conjccture in a very strong form.
As you will see from the Icctutes which will follow our subject is dclinitcly
alive and there are many successes but also unsolvxi problems.
To finish this address 1 just make two remarks. As far as I know this method
was perhaps first used in combinatorial analysis in a paper by Szckercs and Turin
written nearly 50 years ago. They estimate the maximum possible value of an n
by n determinant all whose entries are 0 or 1, and Szele in a paper written more
than 40 years ago estimates the maximum number of possible directed harnilton-
ian cycles of a directed complete graph. Finally, mainly due to considerations
of space I did not discuss our papers with RCnyi on evolution of random graphs
a subject which is certainly alive, RCnyi always wanted to apply these ideas to
physics (change of state) and perhaps to traffic engineering i.e. when a super-
highway at a certain traffic density suddenly turns into a giant parking lot, these
phenomena may be related to the appearance of the giant component if the number
n
of edges is > -( 1 +E). RCnyi’s untimely death prevented his investigating these
2
questions, perhaps this work will be taken up by others in the future.
I would just like to mention one problem due to Shamir which is very attractive
and which R6nyi and I unfortunately missed: Let there be given a set of 3n vertices
and m triples chosen at random. How large must m be that with probability
tending to 1 the m triples should contain n disjoint triples, then of course these
n triples will form a cover. Shamir proved that m<n3I2 suffices for this but perhaps
Welcoming address 5
or even c11 log n triples could suffice. RCnyi and I solved the case of graphs
instead of triples atid there (~j+~l)ong /I suffices.
Most of the results I referred to in this note can be Found in ErdBs-Spencer
probabilistic methodsin combinatoricaanalysjs or in my book Tlir Art of Cowifing,
see also the book of Moon on tournamenls. 13. Rollobis i.4 writing a book on
random graphs which 1 hope will appear soon.
Perhaps I should end by remarking that the probability used here is rather
elementary, in fact, when I lectured 35 years ago at the University of Illinois
on r(n, tI)>C/7 2”12 Doob remnikzd, ‘‘This is very nice but it is morc 1;kecounting
than probability.” I think Doob was right, but perhaps the later applications
are a little more sophisticated.
G. Szekeres and P. Turin, An cxtrenial pl-oblcin in the thcory of dctcrniinants, Math. es
TermCszettudominyi Ertesit8 (1937) 796-806. This was a publication of tlic Hungarian
Academy of Scicnces, thc paper is in Hung:irian.
T. Szelc, Conibinatorial investigations concerning the complete graph, Mat. Fiz. Lapok
50 (1943) 223-256 (in Hungarian).
