ON THE EVOLUTION OF RANDOM GRAPHS by P. ERDŐS and A. RÉNYI. Introduction

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1 ON THE EVOLUTION OF RANDOM GRAPHS by P. ERDŐS ad A. RÉNYI Itroductio Dedicated to Professor P. Turá at his 50th birthday. Our aim is to study the probable structure of a radom graph r N which has give labelled vertices P, P,..., P ad N edges ; we suppose that these N edges are chose at radom amog the l 1 possible edges, so that all ~ = C, possible choices are supposed to be equiprobable. Thus 1V if G,,,,, deotes ay oe of the C,,N graphs formed from give labelled poits ad havig N edges, the probability that the radom graph -P,N is idetical with G,,,N is 1. If A is a property which a graph may or may ot possess, C,N we deote by P N (A) the probability that the radom graph T.,N possesses the property A, i. e. we put P,N (A) = A 'N where A,N deotes the C N umber of those G,N which have the property A. A other equivalet formulatio is the followig : Let us suppose that labelled vertices P,, P,..., P are give. Let us choose at radom a edge amog the l I possible edges, so that all these edges are equiprobable. After this let us choose a other edge amog the remaiig I - 1 edges, ad cotiue this process so that if already k edges are fixed, ay of the remaiig () k edges have equal probabilities to be chose as the ext oe. We shall - study the "evolutio" of such a radom graph if N is icreased. I this ivestigatio we edeavour to fid what is the "typical" structure at a give stage of evolutio (i. e. if N is equal, or asymptotically equal, to a give fuctio N() of ). By a "typical" structure we mea such a structure the probability of which teds to 1 if -* + - whe N = N(). If A is such a property that lim P,N,( ) ( A) = 1, we shall say that almost all" graphs G,N() --- possess this property. 17 A Matematikai Kutató Itézet Közleméye! V. A/1-.

2 18 ERDŐS-RÉNYi The study of the evolutio of graphs leads to rather surprisig results. For a umber of fudametal structural properties A there exists a fuctio A() tedig mootoically to + - for -i- - such that (1) lim P,N() (A) _ -+- i if lim X(v) = 0 - A() if lim N () +co. - A() If such a fuctio A() exists we shall call it a "threshold fuctio" of the property A. I may cases besides (1) it is also true that there exists a probability distributio fuctio F(x) so that if 0 < x < + - ad x is a poit of cotiuity of F(x) the () (3) lim P,N()(A) -.- Clearly (3) implies that i lim ) = x. "M - + m P,N()(A)- F(x) if ->- A() If () holds we shall say that A() is a regular threshold fuctio" for the property A ad call the fuctio F(x) the threshold distributio fuctio of the property A. For certai properties A there exist two fuctios A,() ad A () both tedig mootoically to +- for -->-+-, ad satisfyig lim Á() = 0, - - Al() such that if lim () - A, () _ A () if lim N() - Al() _ a A () (4) lim P,N()(A) _ if lim sup N() < 1 -a+m A l () 1 if liro if +() ~' > ro A,.() If (3) holds we call the pair (Al(), A ()) a pair of sharp threshold"-fuctios of the property A. It follows from (4) that if (A j (), A ()) is a pair of sharp threshold fuctios for the property A the A, () is a (ordiary) threshold fuctio for the property A ad the threshold distributio fuctio figurig i () is the degeerated distributio fuctio jr t (x) _ 1 0 (1 for x < 1 for x>1

3 ON THE EVOLUTION OF RANDOM GRAPHS 1 9 ad covergece i () takes place for every x 1. I some cases besides (3) it is also true that there exists a probability distributio fuctio G(y) defied for -- < y < -)- - such that if y is a poit of cotiuity of G(y) the (ő) lim P.N()(A) = G(y) - if lím NT() - A,() = y -- x A () if (ő) holds we shall say that we have a regular sharp threshold ad shall call G(y) the sharp-threshold distributio fuctio of the property A. Oe of our chief aims will be to determie the threshold respectively sharp threshold fuctios, ad the correspodig distributio fuctios for the most obvious structural properties, e. g. the presece i r N of subgraphs of a give type (trees, cycles of give order, complete subgraphs etc.) further for certai global properties of the graph (coectedess, total umber of coected compoets, etc. ). I a previous paper [7] we have cosidered a special problem of this type ; we have show that deotig by C the property that the graph is coected, the pair C,() - 1 log. C () = is a pair of strog threshold fuctios for the property C, ad the correspodig sharp-threshold distributio fuctio is e -v ; thus we have proved' that puttig A(1) AT = 1 log + y + o(.) we have a (Ó) lim PM)(C) = e_e Y (- - < y < + -) - - I the preset paper we cosider the evolutio of a radom graph i a more systematicc maer ad try to describe the gradual developmet ad step-by-step uravellig of the complex structure of the graph F.,N whe N icreases while is a give large umber. We succeeded i revealig the emergece of certai structural properties of -V N. However a great deal remais to be doe i this field. We shall call i 10. the attetio of the reader to certai usolved problems. It seems to us further that it would be worth while to cosider besides graphs also more complex structures from the same poit of view, i. e. to ivestigate the laws goverig their evolutio i a similar spirit. This may be iterestig ot oly from a purely mathematical poit of view. I fact, the evolutio of graphs may be cosidered as a rather simplified model of the evolutio of certai commuicatio ets (railway, road or electric etwork systems, etc.) of a coutry or some other uit. (Of course, if oe aims at describig such a real situatio, oe should replace the hypothesis of equiprobability of all coectios by some more realistic hypothesis.) It seems plausible that by cosiderig the radom growth of more complicated structures (e. g. structures cosistig of differet sorts of "poits" ad coectios of differet types) oe could obtai fairly reasoable models of more complex real growth processes (e. g. i Partial result o this problem has bee obtaied already i 1939 by P. ERDős ad 11. WHITNEY but their results have ot bee published. *

4 0 ERDÖS-RÉNYI the growth of a complex commuicatio et cosistig of differet types of coectios, ad eve of orgaic structures of livig matter, etc.) cotai the discussio of the presece of certai compoets i a radom graph, while 4-9. ivestigate certai global properties of a radom graph. Most of our ivestigatios deal with the case whe N() - e with c > 0. I fact our results give a clear picture of the evolutio of r N() whe c = N() (which plays i a certai sese the role of time) icreases. I 10. we make some further remarks ad metio some usolved problems. Our ivestigatio belogs to the combiatorical theory of graphs, which has a fairly large literature. The first who eumerated the umber of possible graphs with a give structure was A. CaYLEY [1]. Next the importat paper [] of G. PÓLYA has to be metioed, the startig poit of which were some chemical problems. Amog more recet results we metio the papers of G. E. UHLENBEcK ad G. W. FORD [5] ad E. N. GILBERT [G]. A fairly complete bibliography will be give i a paper of F. HaRARY [8]. I these papers the probabilistic poit of view was ot explicitly emphasized. This has bee doe i the paper [9] of oe of the authors, but the aim of the probabilistic treatmet was there differet : the existece of certai types of graphs has bee show by provig that their probability is positive. Radom trees have bee cosidered i [ 14]. I a recet paper [10] T. L. AusTIN, R. E. FAGEN, W. F. PENNEY ad J. RioRDAN deal with radom graphs from a poit of view similar to ours. The differece betwee the defiitio of a radom graph i [10] ad i the preset paper cosists i that i [10] it is admitted that two poits should be coected by more tha oe edge ("parallel" edges). Thus i [10] it is supposed that after a certai umber of edges have already bee selected, the ext edge to be selected may be ay of the possible edges betwee l 1 the give poits (icludig the edges already selected). Let us deote such a radom graph by I N. The differece betwee the probable properties of r N resp, r,n are i most (but ot i all) cases egligible. The correspodig probabilities are i geeral (if the umber N of edges is ot too large) asymptotically equal. There is a third possible poit of view which is i most cases almost equivalet with these two ; we may suppose that for each pair of give poits it is determied by a chace process whether the edge coectig the two poits should be selected or ot, the probability for selectig ay give edge beig equal to the same umber p > 0, ad the decisios cocerig the differet edges beig completely idepedet. I this case of course the umber of edges is a radom variable, hayig the expectatio (l p ; thus if we wat to obtai by this method a radom graph havig i the mea N edges we have to choose the value of p equal to N. We shall deote such a radom graph by F, N. I may (though ot all) of the problems treated i the preset paper it does ot cause ay essetial differece if we cosider istead of r N the radom graph Q*X,. f~l

5 ON THE EVOLUTION OF RANDOM GRAPHS 1 Comparig the method of the preset paper with that of [10] it should be poited out that our aim is to obtai threshold fuctios resp. distributios, ad thus we are iterested i asymptotic formulae for the probabilities cosidered. Exact formulae are of iterest to us oly so far as they help i determiig the asymptotic behaviour of the probabilities cosidered (which is rarely the case i this field, as the exact formulae are i most cases too complicated). O the other had i [10] the emphasis is o exact formulae resp. o geeratig fuctios. The oly exceptio is the average umber of coected compoets, for the asymptotic evaluatio of which a way is idicated i 5. of [10] ; this questio is however more fully discussed i the preset paper ad our results go beyod that of [10]. -Moreover, we cosider ot oly the umber but also the character of the compoets. Thus for istace we poit out the remarkable chage occurig at N -. If L-' - e with c < 1/ the with probability tedig to 1 for + - all poits except a bouded umber of poits of r,n belog to compoets which are trees, while for N tit with c > 1 this is o loger the case. Further for a fixed value of the average umber of compoets of r,n decreases asymptotically i a liear maer with N, whe N :<, while for N > the formula givig the average umber of compoets is ot liear i N. I what follows we shall make use of the symbols O ad o. As usually li a() = a (b()) (where b() > 0 for = 1,,...~ meas that Mim b()l - 0, -+while a() = 0 (b()) meas that la()j. is bouded. The parameters o b() which the boud of ja() 1 may deped will be idicated if it is ecessary ; b() sometimes we will idicate it by a idex. Thus a() = 0, (b()) meas that ~a()i < K(E) where K(E) is a positive costat depedig o e. We write b() a() -b() to deote that Mim a() = b() We shall use the followig defiitios from the theory of graphs. (For the geeral theory see [3] ad [4].) A fiite o-empty set V of labelled poits P l, Pi..., P ad a set E of differet uordered pairs (P;, Pj) with P1 E V, Pi E V, i j is called a graph ; we deote it sometimes by G = {V, E} ; the umber is called the order (or size) of the graph ; the poits Pl, P,..., P are called the vertices ad the pairs (Pi, Pj) the edges of the graph. Thus we cosider o-orieted fiite graphs without parallel edges ad without sligs. The set E may be empty, thus a collectio of poits (especially a sigle poit) is also a graph. A graph G - {V, E } is called a subgraph of a graph Gl = {Vl, El } if the set of vertices V of G Ms a subset of the set of vertices V, of G l ad the set EZ of edges of G is a subset of the set El of edges of G I.

6 ERDŐS-RÉNYi A sequece of k edges of a graph such that every two cosecutive edges ad oly these have a vertex i commo is called a path of order k. A cyclic sequece of k edges of a graph such that every two cosecutive edges ad oly these have a commo vertex is called a cycle of order k. A graph G is called coected if ay two of its poits belog to a path which is a subgraph of G. A graph is called a tree of order (or size) k if it has k vertices, is coected ad if oe of its subgraphs is a cycle. A tree of order k has evidetly k - 1 edges. A graph is called a complete graph of order ~, if it has k vertices ad edges. Thus i a complete graph of order k ay two poits are coected by a edge. A subgraph G' of a graph G will be called a isolated subgraph if all edges of G oe or both edpoits of which belog to G', belog to G'. A coected isolated subgraph G' of a graph G is called a compoet of G. The umber of poits belogig to a compoet G' of a graph G will be called the size of G'. Two graphs shall be called isomorphic, if there exists a oe-to-oe mappig of the vertices carryig over these graphs ito aother. The graph G shall be called complemetary graph of G if G cosists of the same vertices Pl, P,.... P as G ad of those ad oly those edges (P ;, Pj ) which do ot occur i G. The umber of edges startig from the poit P of a graph G will be called the degree of P i G. A graph G is called a saturated eve graph of type (a, b) if it cosists of a + b poits ad its poits ca be split i two subsets V, ad V cosistig of a resp. b poits, such that G cotais ay edge (P, Q) with P E V, ad Q E V ad o other edge. A graph is called plaar, if it ca be draw o the plae so that o two of its edges itersect. We itroduce further the followig defiitios : If a graph G has vertices ad N edges, we call the umber N the "degree" of the graph. V (As a matter of fact 1ti is the average degree of the vertices of G.) If a graph G has the property that G has o subgraph havig a larger degree tha G itself, we call G a balaced graph. We deote by P (... ) the probability- of the evet i the brackets, by M(~) resp. D (á) the mea value resp. variace of the radom variable s. I cases whe it is ot clear from the cotext i which probability space the probabilities or respectively the mea values ad variaces are to be uderstood, this will be explicitly idicated. Especially M,N resp. D,N will deote the mea value resp. variace calculated with respect to the probabilities P.N

7 ON THE EVOLUTION OF RANDOM GRAPHS 3 (7) We shall ofte use the followig elemetary asymptotic formula : i! k e k'- 0 6= valid for k = o('l=). k Our thaks are due to T. GALLAI for his valuable remarks. 1. Thresholds for subgraphs of give type If N is very small compared with, amely if N - o (V) the it is very probable that r.n is a collectio of isolated poits ad isolated edges, i. e. that o two edges of r,n have a poit i commo. As a matter of fact the probability that at least two edges of r,n shall have a poit i commo is by (7) clearly N) 1-~. ( =0 (1T lj 1' Y1 If however íl c I/I where c > 0 is a costat ot depedig o., the the appearace of trees of order 3 will have a probability which teds to a positive limit for -* + -, but the appearace of a coected compoet cosistig of more tha 3 poits will be still very improbable. If N is icreased while is fixed, the situatio will chage oly if N reaches the order of magitude of j3. The trees of order 4 (but ot of higher order) will appear with a probability ot tedig to 0. I geeral, the threshold fuctio for the presece k- of trees of order k is k -1 (k = 3, 4,... ). This result is cotaied i the followig Theorem 1. Let k >_ ad l (k - 1 < Z < _ l 9 lk I be positive itegers. Let k deote a arbitrary ot empty class of coected balaced graphs cosistig of k poits ad l edges. The threshold fuctio for the property that the radom graph cosidered should cotai at least oe subgraph isomorphic with some ele- - k met of y k,, is t. The followig special cases are worth metioig Corollary 1. The threshold fuctio for the property that the radom graph k- cotais a subgraph which is a tree of order k is k - I (k: = 3, ). Corollary. The threshold fuctio for the property that a graph cotais a coected subgraph cosistig of k >_ 3 poits ad k edges (i. e. cotaiig exactly oe cycle) is, for each value of k. Corollary 3. The threshold fuctio for the property that a graph cotais a cycle of order k is, for each value of k _>_. 3.

8 4 ERDŐS-RÉNYI Corollary 4. The threshold fuctio for the property that a graph cotais - 1 ) a complete subgraph of order k >_ 3 is z (i k-1,. Corollary 5. The threshold fuctio for the property that a graph cotais a saturated eve subgraph of type (a, b) (i. e. a subgraph cosistig of a + b a+b poits P1,..., Pa, Q I,... Q b ad of the ab edges (P1, Qj ) is ab. To deduce these Corollaries oe has oly to verify that all 5 types of graphs figurig i Corollaries 1-5. are balaced, which is easily see. Proof of Theorem 1. Let Bk I >_ 1 deote the umber of graphs belogig to the class which ca be formed from k give labelled poits. Clearly if P,N (&k,l) deotes the probability that the radom graph r,n cotais at least oe subgraph isomorphic with some elemet of the class ^Vjk,l, the P,N(&k,l) < k Bk,1 7- l = o ( ` 1-k I J 1 N As a matter of fact if we select k poits (which ca be doe i 171 differet ways) ad form from them a graph isomorphic with some elemet of the class 66k,í (which ca be doe i B k,l differet ways) the the umber of graphs G,N which cotai the selected graph as a subgraph is equal to the umber of ways the remaiig N-1 edges ca be selected from the I other - l possible edges. (Of course those graphs, which cotai more subgraphs isomorphic with some elemet of are couted more tha oce.) k,1 k Now clearly if N = o(_ 1 ) the by P,N( k,l) = O(1) which proves the first part of the assertio of Theorem 1. To prove the secod part of the theorem let -Vki deote the set of all subgraphs of the complete graph cosistig of poits, isomorphic with some elemet of 56 k 1. To ay SE& let us associate a radom variable e(s) such that E(S) = 1 or e(s) = 0 accordig to whether S is a subgraph of P N or ot. The clearly (we write i what follows for the sake of brevity W istead of M,N) (1.) M (~ E(S)1 = M(E(S)) _ () Bk l N - l, Bk,l ( N)'. () () k k I 1-k SE, ik 1 S E.9 ik,1 ~9 l

9 ON THE EVOLUTION OF RANDOM GRAPHS 5 O the other had if SI ad S are two elemets of Z01)1 ad if Sl ad 8 do ot cotai a commo edge the fj -l M(E(SI) E(S» = N-1,~ If S,_ ad S cotai exactly s commo poits ad r commo edges (1 < r < l -1) we have ~ N, -1+r j l M(s(Sl ) E(S)) -. N r - N1-r 0 f41-r (l N O the other had the itersectio of S I ad S beig a subgraph of S I (ad S ) by our suppositio that each S is balaced, we obtai r < l i. e. s z rk S - k l ad thus the umber of such pairs of subgraphs S I ad S does ot exceed Thus we obtai k k -k k _rk 0 B `k kf k-~~ 1. Jz i SE M (~E M(e(S)) +! Bk,1 k!( - k)! i k, 1 ) E(S)~~) k,1-1 ~' N 1 l 1 ;}T k (- 1 +OI N1 J N r ~ Now clearly I 1-1 -l ~ lj! N -1 N-1 < k! ( - k)! ' ~k ~I ~1 1 - k

10 6 ERDŐS--RÉNYI If we suppose that it follows that we have (1.4) I\\\ (Z r SE&( D 1 1 SEM () e(s}) =0 ( k,1 M(8 (8» CU It follows by the iequality of Chebysheff that ad thus P,N Í Z e(s) - Z M( (S)) > 1. M(e(S» SE ffi k,1 SEM k,1 setik,1 (1.5) P,N L e(s) < 1 M(E(S») =O SEM (r' ), SE Y~kj l1 GJ As clearly by (1.) if w -~ + the M(e(S)) + - it follows ot oly that the probability that 1' cotais at least oe subgraph isomorphic with a elemet of k,i teds to 1, but also that with probability tedig to 1 the umber of subgraphs of F N isomorphic to some elemet of ti 1 ; 1 will ted to +O with the same order of magitude as GJ 1. Thus Theorem 1 is proved. It is iterestig to compare the thresholds for the appearace of a subgraph of a certai type i the above sese with probability ear to 1, with the umber of edges which is eeded i order that the graph should have ecessarily a subgraph of the give type. Such -compulsory" thresholds have bee cosidered by P. TURÁN [11] (see also [1]) ad later by P. ERDŐS ad A. 11. STONE [ 17 ]). For istace for a tree of order k clearly the compulsory - ] threshold is I (k ) + 1 ; for the presece of at least oe cycle the compulsory threshold is while accordig to a theorem of P. TURÁN [11] for complete subgraphs of order k the compulsory threshold is (k?) ( - r ) + (k - 1) -v where r=-- (k-- 1) ~- I the paper [13] of T. KŐVÁ RI, ~1 Is - 11' V. T. Sós ad P. TuRÁN it has bee show that the compulsory threshold for the presece of a saturated eve subgraph of type (a, a) is of order of magitude ot greater tha a. I all cases the "compulsory" thresholds i TURÁN's sese are of greater order of magitude as our "probable" thresholds. k,1 I.

11 O\ THE EVOLUTION OF RANDOM GRAPHS 7. Trees Now let us tur to the determiatio of threshold distributio fuctios for trees of a give order. We shall prove somewhat more, amely that if k- N e k -1 where o > 0, the the umber of trees of order k cotaied i r,n has i the limit for -y + oo a Poisso distributio with mea value _ ( P )k 1 kk-z. This implies that the threshold distributio fuctio for V trees of order k is 1 - e - ~. I provig this we shall cout oly isolated trees of order k i F N, i- e. trees of order k which are isolated subgraphs of I' N' Accordig to Theorem 1. this makes o essetial differece, because if there would be a tree of order k which is a subgraph but ot a isolated subgraph of P N, the rr,,n would have a coected subgraph cosistig of k + 1 poits ad the probability- ( of this is tedig to 0 if N = o l -. k which coditio is fulfilled i our k- g. case as we suppose N - k-i. Thus we prove Theorem a. If lim N( k- = e > 0 ad -r k deotes the umber of isolated -~ k-1 trees of order k i Fr, N() the A 1 (.1) p, e- ~ "M ti'()(t1 --, - Í ) -- or j = 0, 1,. where - ( Q )k-1 kk- A k! For the proof «e eed the followig Lemma 1. Let E 1' eg- ' Er,z be sets of radom variables o some probability space ; suppose that E (1 < i <. l ) takes o oly the values 1 ad 0. If j'. hm M (-'i, Ei,... Ei,) -, r! uiformly i. r for r = 1,,..,, where A > 0 ad the summatio is exteded over all combiatios (i l, i,..., ir ) of order r of the itegers 1,,..., l, the r dje - x (.4) lim p ~~ -' i = j l = (j = 0, 1,... ) i=1 j! i. e. the distributio of the sumpoisso- ER, teds for -~ + co to the distri-butio with mea value ~.

12 8 E RDŐS-RÉN YI Proof of Lemma 1. Let us put (.5) Clearly ( :6) M 1Si,<iz<...<i.<1 thus it follows from (.3) that P11(j) = P ~Z Ee =! I. i-t (.7 ) lim Z p, (j) >+mj=r l71 r ri uiformly i r. It follows that for ay z with I z < ~ r (.8) hm 'Y P(9) I I j ~ z r = ~, ( z ()r = e' _~ - 1 yt' r=1 j-r l r Í.r! But j r (r = 1,, ) (.9),Z ( ~P(j) r1) Zr -_.G P(7) (1 + z)j - 1. r-1 j=r l j=o Thus choosig z = x - 1 with 0 < x < 1 it follows that ('.10) lim > P(?) j = e~ (x - ' ) - --m j=o for 0<x< 1. It follows easily that (.10) holds for x = 0 too. As a matter of fact puttig G (x) xi, we have for 0 < x < 1 j=o P,(O) - e -1 I < I G,(x) - I + I G,,( :x) - P ' (O) I + I e'(x-1) - e-d I. _1s however ad similarly it follows that Thus we have G (x) - P'(0) I - x P(l ) x j --t ef.(x-1) - e- I < x I P(O) - e -. I < I G'(x) - e4x-1) I - x. lim sup I P,(O) -- e - ' I< x ; as however x > 0 may be chose arbitrarily small it follows that lim P, (0) = e -

13 ON THE EVOLUTION OF RANDOM GRAPHS 9 i. e. that (.10) holds for x = 0 too. It follows by a well-kow argumet that (.11) lira P,(j) _ -, (j = ) f As a matter of fact, as (.10) is valid for x = 0, (.11) holds for j = 0. If (.11) is already proved for j < s - 1 the it follows from (.10) that (.1) lim P,(j) xi - S = xi - S for 0<x< i=5 Í=S ~~ By the same argumet as used i coectio with (.10) we obtai that (.1) holds for x = 0 too. Substitutig x = 0 ito (.1) we obtai that (.11) holds for j = s too. Thus (.11) is proved by iductio ad the assertio of Lemma 1 follows. Proof of Theorem a. Let Tk) deote the set of all trees of order k which are subgraphs of the complete graph havig the vertices P i, P,... 5 P,,. If SETk") let the radom variable e(s) be equal to 1 if S is a isolated subgraph of r, N ; otherwise e(s) shall be equal to 0. We shall show that the coditios of Lemma 1 are satisfied for the sum _Y e(s) provided that N= N() - SETk) k- ek -1 ad A is defied by (.). As a matter of fact we have for ay SETk) ( - k~ I 1 (.13) M(E(S)) _ N +1 = ( lk-1 e _N' 1 +0 It J N More geerally if S1, 8,..., Sr (Si E T(R) ) have pairwise o poit i commo the clearly we have for each fixed k >_ - 1 ad r -> 1 provided that a+oo, N-;+O / I Ū M("(800) e S )~ = IV - r(k - 1) j,(k-1)r - Nrk r l (.14)... ( r - e ( 1 O( ~r, 1 (l! N where the boud of the 0 term depeds oly o k. If however the Si ( = = 1,,..., r) are ot pairwise disjoit, we have rk I (.15) M (e (Si) e(s)... e(s,» = 0.

14 3 0 ERDŐS--RÉNYI Takig ito accout that accordig to a classical formula Of CAYLEY (l] the umber of differet trees which ca be formed from k labelled poits is equal to kk -, it follows that kk- rkr ~NII-k-' _ Nrk y, (.16) M(E(Sl) E(S )... E(S,)) _ -- e (1 +0 ~~ k! r! where the summatio o the left had side is exteded over all r-tuples of trees belogig to the set T(k) ad the boud of the 0-term depeds oly o k. Note that (.16) is valid idepedetly of how N is tedig to +-. This will be eeded i the proof of Theorem 3. Thus we have, uiformly i r 7! r (.17) lim _ M(E(S1 ) e(s )... E(S,,» = k-s -ip k-i for r = 1,,... where A is defied by (.). Thus our Lemma 1 ca be applied ; as z k = e(s) Theorem is proved. SET k) We add some remarks o the formula, resultig from (.16) for r = 1 (N _ N,k I k;k - (.18) M(Tk) _ ja e i+o l N k! ~ 1~ Let us ivestigate the fuctios kk- tk-1 C -kf mk (t) _ (k= 1,,... ). accordk! ig to (.18) mk i - I is asymptotically equal to the average umber of trees of 1 order k i r,n. For a fixed value of k, cosidered as a fuctio of t, the value of mk (t) icreases for t < L - 1 ad decreases for t > k - 1 ; thus for a fixed k k value of the average umber of trees of order k reaches its maximum for N ; k the value of this maximum is 1,k-1 'k- (1 - e-(k-i) ` -~ k. For large values of k we have evidetly M,*~ r k! V 7c k51

15 ON THE EVOLUTION OF RANDOM GRAPHS 3 1 It is easy to see that for ay t > 0 we have Mk(t) >_ mk+>_(t) (k = 1,,... ). The fuctios y = m k(t) are show o Fig. 1. It is atural to ask what will happe with the umber r k of isolated trees of order k cotaied i I' N if N() --*+ -. As the Poisso distributio k-1 (Ij ~ e-f ~ l~ is approachig the ormal distributio if ~. + -, oe ca guess that r k will be approximately ormally distributed. This is i fact true, ad is expressed by Z Fi.ure la. e o x (c) t i c F gure 1b.

16 3 (.19) Theorem h. If ERDŐS-RÉNYI N() k-~ + 00 k-1 but at the same time N() - 1 log -- k- l loglog (.0) lim k k o -+- the deotig by "'k the umber of disjoit trees of order k cotaied as subgraphs i I' N( ) (k = 1,,... ), we have for - - < x < + - {.1) hm P.M.) ~ Zk - -3I,N() <X l = O (X) -+- l Vm,N() where (.) ad kk - (N~k-1 _ kn I,N - e k! (.3) <P(x) = 1 x e - du. V Proof of Theorem h. Note first that the two coditios (.19) ad (.0) are equivalet to the sigle coditio lim M N()= + ad as -y+ m M ( c k ) ti 1VI,N this meas that the assertio of Theorem b ca be expressed by sayig that the umber of isolated trees of order k is asymptotically ormally distributed always if ad N ted to +- so, that the average umber of such trees is also tedig to +-. Let us cosider Now we have evidetly, usig (.16) M(, r k) = N1(s Z () E( S )) r ) r ~rjv) -r r! -1,N M (~k) _ ~ 1-F l hl! h!... hj! f1 k where M,N is defied by (.). Now as well kow (see [16], p. 176) (.4) 1 r! h l! h!... h ] -! i~lrg=r, hi>1 - ag) r

17 ON THE EVOLUTION OF RANDOM GRAPHS 33 where v(í) are the Stirlig umbers of the secod kid (see e. g. [16], p. 168) defied by r (.5) xr = 60) x(x - 1)... (x - Í=i Thus we obtai (.6) M (Tk) _ (1 + 0 Ir~ 09) 1I,N. Í=1 Now as well kow (see e. g. [16], p. 0) + + xr W Aj xr r (.7) e a(ex 1) - 1 = orq) - d (1)Ai Í=1 r=í r~ r=1 91.J=1 Thus it follows that (,8) r dr l + a ~k x, 6rÍ)1 = -- e.(e -1) I = e- Í=1 dxr x=0 k=0k We obtai therefrom T, V~ - M""" ~ ~ I,N M r r N (.9) M ~ ( e ry (k - f 11T í I MroN) ~ (1 + 0 ~- ~ ~.,N,N k=0 S k +. k Now evidetly ki e -1 (k - ~t}r is the r-th cetral momet of the Poisso distributio with mea value. It ca be however easily verified that the momets of the Poisso distributio appropriately ormalized ted to the correspodig momets of the ormal distributio, i. e. we have for r = 1,,... + ~ x a Ak e (.30) lim ( f. - a )rl = xr e dx. ).y+. k i k. V 1 I view of (.9) this implies the assertio of Theorem b. I the case N () =1 log + k- 1 loglog + y + o() whe k k the average umber of isolated trees of order k i I',N() is agai fiite, the followig theorem is valid. Theorem c. Let rk deote the umber of isolated trees of order k i (k=1,,,.. ). The if l',n (.31) í17()= Ilog+ k. k I loglog+y+o() where - - < y < + -, we have ~ (.3) "l Í ~ P,N() (""k = i) _ +- j. where e -ky (.33) A = k k ' (j = 0, 1,... ) 3 A Matematikai hutató Itézet Közleméyei V. A(1--.

18 3 4 ERDÖS---RÉNYI Proof of Theorem c. It is easily see that uder the coditios of Theorem c hm M.N() (Tk) - - Similarly from (.16) it follows that for r = 1,,... hm ~ M.N() ~~(sl) E(~)... E(~r)) _ SIET k() ad the proof of Theorem c is completed by the use of our Lemma 1 exactly as i the proof of Theorem a. Note that Theorem c geeralizes the results of the paper [7], where oly the case 1c = 1 is cosidered. 3. Cycles Let us cosider ow the threshold fuctio of cycles of a give order. The situatio is described by the followig Theorem 3a. Suppose that (3.1) N() -c where e > 0.. Let yk deote the umber of cycles of order k cotaied i F,,N (k= 3, 4,... ). The we have (3.) A1 e -a "m P.N() (Yk om? 0, 1,...) where (3.3) ` ( c)k. k Thus the threshold distributio correspodig to the threshold fuctio A() = for the property that the graph cotais a cycle of order k is 1 - e- k ( 1)k It is iterestig to compare Theorem 3a with the followig two theorems : Theorem 3b. Suppose agai that (3.1) holds. Let yk deote the umber of isolated cycles of order k cotaied i r N (k = 3, 4,... ). The we have (3.4) lira P,N() (Yk * = j) _ UJ e u -+? (Í = 0, 1,.. ) where (3.5) ( r e-c) k,u = k Remark. Note that accordig to Theorem 3b for isolated cycles there does ot exist a threshold i the ordiary sese, as 1 - e-11 reaches its maximum 1 - e kek for c = i. e. for N() -~ ad the agai decreases ; ~r r'

19 ON THE EVOLUTION OF RANDOM GRAPHS 35 thus the probability that r N cotais a isolated cycle of order k ever approaches 1. Theorem 3c. Let 8 k deote the umber of compoets of -P.,N cosistig of k > 3 poits ad k edges. If (3.1) holds the we have (3.6) where "M P,N() (ak = Í) _ w1 (9=0,1,...) (3.7) w _ ( C e-c)k (1 + k + Z } k-3 ~ k t! (k-3)!. Proof of Theorems 3a., 3h. ad 3e. As from k give poits oe ca form (k - 1)! cycles of order k we have evidetly for fixed k ad for N- 0() M (Yk) = 1 (, k, 1 (k - 1)! ~ ) -k N,1 N-k. L ~ while M(fh)= 1 ~ ~( k - -1)! I ív e -N l k k As regards Theorem 3c it is kow (see [10] ad [15]) that the umber of coected graphs Gk.k (i" e. the umber of coected graphs cosistig of k labelled vertices ad k edges) is exactly (3.10) () k = (k - 1)! I + k (kkk 3) j. Now we have clearly (3.11) M (S k) = I lti k} I ;~ ~, N k 1 8 k kk-3 - k ' ~ I - + k k! (k - 3)!, 3*

20 -RÉNYI 36 For large values of k we have (see [15]) (3.1) ad thus (3.13) ~? kk 8 N i-z ~k ~- ee M (k) -, 4 k --. For íl' we obtai by some elemetary computatio usig (7) that for large values of k (such that k = o (3-4). (3.14) M(bk) (4.1) P,N (T) I lk k3 e 4k Usig (3.8), (3.9) ad (3.11) the proofs of Theorems 3a, 3b ad 3c follow the same lies as that of Theorem a, usig Lemma 1. The details may be left to the reader. Similar results ca be proved for other types of subgraphs, e. g. complete subgraphs of a give order. As however these results ad their proofs have the same patter as those give above we do ot dwell o the subject ay loger ad pass to ivestigate global properties of the radom graph T,,,. 4. The total umber of poits belogig to trees We begi by provig Theorem 4a. If N = o() the graph r N is, with probability tedig to 1 for ->- +-, the uio of disjoit trees. Proof of Theorem 4a. A graph cosists of disjoit trees if ad oly if there are o cycles i the graph. The umber of graphs G,N which cotai at least oe cycle ca be eumerated as was show i 1 for each value k o the legth of this cycle. I this way, deotig by T the property that the graph is a uio of disjoit trees, ad by T the opposite of this property, i. e. that the graph cotais at least oe cycle, we have. () - k.n-k,. _ O ~Nj. It follows that if N = o() we have lim P N(T) = 1 which proves Theorem 4a. - - If N is of the same order of magitude as i, e. il' - c with c > 0, the the assertio of Theorem 4a is o loger true. Nevertheless if c < 1/, i ) N)

21 ON THE EVOLUTION OF RANDOM GRAPHS 37 still almost all poits (i fact - 0(1) poits) of I' N belog to isolated trees. There is however a surprisigly abrupt chage i the structure of ",N with 1V - e, whe c surpasses the value. If c > 1/ i the average oly a positive fractio of all poits Of r,n belog to isolated trees, ad the value of this fractio teds to 0 for c +. Thus we shall prove Theorem 4b. Let ti',,, deote the umber of those poits of T N which belog to a isolated tree cotaied i h,n. Let us suppose that N(v) (4.) hill = c > The we have (/ 1 for c < 1/ M ( y ~.N()) _ 1 (4.3) lim - x(c) 1 -. ;_ T for c > - c where x = x(c) is the oly root satisfyig 0 < x < 1 of the equatio (4.4) x e - X= tee-c, which ca also be obtaied as the sum of a series as follows : kk (4.5) x(c) _ 1 ( c e c) k k=1 V Proof of Theorem 4b. 'e shall eed the well kow fact that the iverse fuctio of the fuctio (4.6) y = x e - X (0 :< X :< 1) has the power series expasio, coverget for 0 < y < 1 e +. kk-i y k (4.7) x = `' k=1 kf Let T k deote the umber of isolated trees of order k cotaied i -r.,n. The clearly (4.8) V N = N k -r d k=1 ad thus (4.9) M(i',N) _ ` k M(T I ). k=1 By (.18), if (4.) holds, we have _ 1 V1 (4.10) lim 1 M k- (~k) ( c e -c)k -+. c ki k

22 3 8 ERDŐS-RÉNYI Thus we obtai from (4.10) that for c < 1/ (4.11) lim if M(V,N()) > I kk-1( c e-c)k for ay s>.1. J-+~ - c k_ 1 k! As (4.11) holds for ay s >_ 1 we obtai (4.1) lim if M(V,N()) > 1 kk -1 ( c e -c)k -+- C k! But accordig to (4.7) for c < 1j we have kk- 1 ( c e-c) k =c. k=1 k! Thus it follows from (4.1) that for c <_ 11 (4.13) lim if M(V,N()) >_ As however V,N() ad thus lim sup ~I(yr,N()) < 1 it follows that -~ if (4.) holds ad c < 1/ we have (4.14) lim M (V,N()) = I. ->+. Now let us cosider the case c > 1. It follows from (.18) that if (4.) holds with c > 1/ we obtai (4.15) M (V N()) = kk -1 N() - N)~ k e 0(1) N k! where the boud of the term 0(1) depeds oly o c. As however for N() N - c with c > 1/ it follows that kk -1 J AT() e - N(), k k=,,+1 k! (4.16) M(V,N()) = x (l )1 +0(1) _ where x = x (N() 1 is the oly solutio with o < x < 1 of the equatio I` J I N() - N() xe - x = e Thus it follows that if (4.) holds with c > 1/ we have M(V,N( )) _ x(c) (4.17) lim -.+m c where x(c) is defied by (4.5).

23 ON THE EVOLUTION OF RANDOM GRAPHS 3 9 The graph of the fuctio x(c) by Fig. 11). The fuctio is show o Fig. la ; its meaig is show for c < 11 for c > 11 is show o Fig. a. Figure a. Y Gfc1-1- c) 0 Figure b.

24 40 ERDŐS-RÉNYI Thus the proof of Theorem 41) is complete. Let us remark that i the same way as we obtaied (4.16) we get that if (4.) holds with c < 1/ we have (4.18) M( V,N()) = - 0(1) where the boud of the 0(1) term depeds oly o c. (However (4.18) true for c = 1 as will be show below.) It follows by the well kow iequality of Markov is ot, (4.19) P(~ > a) _<- 1 M(s) a, valid for ay oegative radom variable ~ ad ay a > M(~), that the followig theorem holds : Theorem 4c. Let "7,N deote the umber of those poits of r,n which belog to isolated trees cotaied i F v. The if w teds arbitrarily slowly to +- for --)- +- ad if (4.) holds with c < 11 we have (4.0) lim P(VM) > = The case c > 1 f is somewhat more ivolved. We prove Theorem 4d. Let ",,,N deote the umber of those poits of r,n which belog to a isolated tree cotaied i. Let us suppose that (4.) holds with c > 1/. It follows that if w teds arbitrarily slowly to +we have (4.1) "M P 1,,N() - 'rl() V 9 N() x I ( 1 ~ a) where x = x ~ N()-~ is the oly solutio with 0 < x < 1 of the equatio l Te -x ~4T(} - _e. N() k Proof. We have clearly, as the series } k ( ce -c)k is coverget, h`_, k t D (V,N()) = 0(). Thus (4.1) follows by the iequality of Chebyshev. Remark. It follows from (4.1) that we have for ay c > 11 ad ay e > o (4.) hm P ivmn() _ x(c) < e = c where x(c) is defied by (4.5). As regards the case c ij we formulate the theorem which will be eeded latter.

25 ON THE EVOLUTION OF RANDOM GRAPHS 41 Theorem 4e. Let V, N (r) deote the umber of those poits of F,,N which belog to isolated trees of order >_ r ad r v (r) the umber of isolated trees of order >_ r cotaied i F?i N. If N() we have for ay 6 > o (4.3) ad (4.4) lim P,N()(r) 1 ~ k k 1 e - k < = 1 --x. - k! hm Pl --. Z,N()(7) q kk- - ~ 6-h " k/r k! The proof follows the same lies as those of the precedig theorems. 5. The total umber of poits belogig to cycles Let us determie first the average umber of all cycles i r N. W e prove that this umber remais bouded if N() c ad c < 1' but ot if c =1f. Theorem 5a. Let H,N deote the umber of all cycles cotaied i F,,,N. The we have if N() -V e holds with c < 1 (5.1) 'I'll Ta M (H, N ()) = log 1 1 c - c - c while we have for c - 1 (5.) M(H,,N()) - 4 log. -r,,,n Proof. Clearly- i y k is the umber of all cycles of order I cotaied i we have H,N = G ykk=i Now (5.1) follows easily, takig ito accout that (see (3.8)) M (Yk) ll k N k (k~ (k ) k + ~ N 1 0 k

26 4 ERDŐS--RÉNYI If c = 1f we have by (3.8) 30 (5.1) M (Yk) Ic e 1 _ 3k' 1 As } e -loge, it follows that (5.) holds. Thus Theorem 5a k=3 k 4 is proved. Let us remark that it follows from (5.) that (4.18) is ot true for c = 1 /. Similarly as before we ca prove correspodig results cocerig the radom variable H N itself. We have for istace i the case c = 1/ for ay s > o (5.5) lim P ( H.N() - 1 ' < a =1. -.+~ log 4 This ca be proved by the sae method as used above : estimatig the variace ad usig the iequality of Chebyshev. A other related result, throwig more light o the appearace of cycles i P,N rus as follows. Theorem 5h. Let K deote the property that a. graph cotais at least oe cycle. The we have if. N() o holds with c 1 / (5.6) lira P,N()( K ) = 1-1- c e+'~y. --+ z Thus for e = 1 it is,almost sure" that r N() cotais at least oe cycle, while for c < 1 the limit for -> + co of the probability of this is less tha 1. Proof. Let us suppose first c <. By a obvious sieve (takig ito accout that accordig to Theorem 1 the probability that there will be i I',N() with N() ~- o (c < 1/) two circles havig a poit i commo is egligibly small) we obtai (5.7) _ lim M(H,~v( i) lim P,N()(K ) = e 1- e ec+~~. - Thus (5.6) follows for c < 1/. As for c 1 / the fuctio o the right of (5.6) teds to 1, it follows that (5.6) holds for c = 1/ too. The fuctio y = V1 - c ec+c = is show o Fig. 3. We prove ow the followig Theorem 5c. Let H, N deote the total umber of poits of rn which belog to some cycle. The we have for N = N() o with 0 e c <' 1/ 4 (5.8) lim M c a (H,N()) -.+~ 1-c

27 ON THE EVOLUTION OF RANDOM GRAPHS 43 1 tj=1 - v1-r,.ec, c 0 Figure 3. 1 Proof of Theorem 5c. As accordig to Theorem 1 the probability that two cycles should have a poit i commo is egligibly small, we have by (5.3) M (H,N()) (c) 3 4c 3 k y4_ Zk= t (1 -c) 1-c The size of that part of I',N which does ot cosist of trees is still more clearly show by the followig Theorem 5d. Let ~0,N deote the umber of those poits of r N which belog to compoets cotaiig exactly oe cycle. The we have for N = N() - e i case c * 1 1 (5.9) lie M(O,N()) = 1 ( ee- c) k ~1 + k while for c =1/ (5.10) 0-3 -~ g g 1! 1 (k -3)! we have h ~ 1 ~ M 3,r3 (,N() 1 where I(x) deotes the gamma-fuctio 1'(x) _ f tx- 1 e-t dt for x > 0. 0

28 4 4 ERDŐS-RÉNYI Proof of Theorem 5d. (5.9) follows immediately from (3.11) ; for c- 1 ; we have by (3.14) M ('9i ;,N()) k' ~I 1 3 ) e ~.ti b 3. 1 Remark. Note that for c -~ 1/ 1 (J ce -c ) k kk -3 1 j + 1k, +... (k - 3)11 4(1 - c) Thus the average umber of poits belogig to compoets cotaiig exactly oe cycle teds to + - as 1 for c -* 1 1`. 4(1 - c) We ow prove Theorem 5e. For í1() -v c with 0 < c < '/ all campoets of r.ly() either trees or compoets cotai- are with probability tedig to 1 for ig exactly oe cycle. Proof. Let ip,,n, deote the umber of poits of F,, v; belogig to compoets which cotai more e dges tha vertices ad the ' umber of vertices of which is less tha V log. We have clearly for X() c with c < 1' M (V,N()) <_ [ log -k k I 1 -~-~ ( - 1~ ( kl r log k=4 1 l k (1 Thus P(V,N() > 1) - 0( 1 log) O the other had by Theorem 4c the probability that a compoet cosistig of more tha V log poits should ot be a tree teds to 0. Thus the assertio of Theorem 5e follows. 6. The umber of compoets Let us tur ow to the ivestigatio of the average umber of compoets of It will be see that the above discussio cotais a fairly complete solutio of this questio. We prove the followig

29 ON THE EVOLUTION' OF RANDOM GRAPHS 45 Theorem 6. If ~ N deotes the umber of compoets of r,, N the we have if N() - o holds with 0 < c < 1 (6.1) M (s,n()) = 9t - N() -}- 0(1) where the boud of the 0-term depeds oly o c. If N() (6.) M(,N()) = - N() + O(log ). we have I f N() -o holds with c > 1 we have lim M(~rz,N()) X(C) _ - - } I x(e) --+m 9t C where x =x(c) is the oly solutio satisfyig 0 < x < 1 of the equatio = Zee -c, i. e. xe - z = x(c) _ kk-1 ( CE c)is k=1 k! Proof of Theorem 6. Let us cosider first the case C < 1. Clearly if we add a ew edge to a graph, the either this edge coects two poits belogig to differet compoets, i which case the umber of compoets is decreased by 1, or it coects two poits belogig to the same compoet i which case the umber of compoets does ot chage but at least oe ew cycle is created. Thus (6.5) S,N - ( - N) < H,N where H,N is the total umber of cycles i '.,N. Thus by Theorem 5a it follows that (6.1) holds. Similarly (6.) follows also from Theorem 5a. Now we cosider the case 1 C > -. It is easy to see that for o < y < 1 we have (see e. g. [14]) e I,-.y k x (6.6) k=1 - k! = x - - where + b k k-1 y k k=1 k! I fact accordig to a well kow theorem of the theory of graphs (see [4], p. 9) beig a geeralizatio of Euler's theorem o polyhedra we have N - + ~,N = = x,n, where x,n - the,cyclomatic umber" of the graph r,n - is equal to the maximal umber of idepedet cycles, i I',N (For a defiitio of idepedet cycles see [4] p. 8).

30 4 6 ERDŐS-RÉNYI x ca be characterized also as the oly solutio satisfyig 0 < x < 1 of the equatio xe_t = y. - It follows that if 1V(i?) c holds with c <'/ we have (6.8) MN()) N() 4 () ~ _ - X - -{- 0(1) = - 1' () -f- 0(1) N() which leads to a secod proof of the first part of Theorem 6. To prove the secod part, let us remark first that the umber of compoets of order greater tha 4 is clearly <. Thus if,ti.(a) deotes the umber of compoets of order < 4 of F,N We have clearly (6.9) M(',N) = M(~,N( 4 )) -I- O fs A The average umber of compoets of fixed order k which cotai?~ I k at least k edges will be clearly- accordig to Theorem 1 of order f. e. bouded for each fixed value of lc. As 4 ca be chose arbitrarily large we obtai from (6.9) that (6.10) MGl?,,) - o, M(Tk) - k=1 Accordig to (.18) it follows that - ` kk- N - k (6.11) M( ;,N) e.k 1 k! ad thus, accordig to (6.6) if N() - c holds with c > 1,1 We have (6.1) lim MG,N(1) = 1. x(c) - xu(c) -- e where x(c) is defied by (6.4). Thus Theorem 6 is completely proved. Let us add some remarks. Theorem 6 illustrates also the fudametal chage i the structure of which takes place if IV passes. While the III average umber of compoets of -r,,,n (as a fuctio of N with fixed) decreases liearly if N < this is o loger true for N > ; the average umber of compoets decreases from this poit oward more ad more slowly. The graph of 1-c for 0 _<c<_ -1- ( 5 (6.13) w(c) = lit M,N()) N() 1 x(c) ' 1 C fx(c) - for c > / - 1

31 ON THE EVOLUTION OF RANDOM GRAPHS 4 7 as a fuctio of c is show by Fig. 4. From Theorem 6 oe ca deduce easily that i case N() - c with c < 1/ we have for ay sequece w tedig arbitrarily slowly to ifiity (6.14) "M P(I S,N() - -{- N() I < Co) = 1 (6.14) follows easily by remarkig that clearly `te >_ : - N. Z z=zfc)= I-Cfor0:cs% c (x (c)- x cl) for c > % Ő 1 1 Figure 4. -c For the case N() - c. with c >_ 1/ oe obtais by estimatig the variace of ~,N() ad usig the iequality of Chebyshev that for ay e > 0 (6.15) lim P S.N() _ I fx(c - x_ (e < e l , c ~ J The proof is similar to that of (4.1) ad therefore we do ot go ito details. 7. The size of the greatest tree If N - c with c < 1 / the as we have see i 6 all but a fiite umber of poits of F, N belog to compoets which are trees. Thus i this case the problem of determiig the size of the largest compoet of F,, N reduces to the easier questio of determiig the greatest tree i ",,N. This questio is aswered by the followig. Theorem 7a. Let do,~ deote the umber of poits of the greatest tree which is a compoet of F., N* Suppose N = N() - c with c + 1/. Let w be a sequece

32 4 8 ERDŐS-RÉ NYI tedig arbitrarily slowly to + ~. The we have (7.1) slim P 4,N() > f log - loglog I + (or, = 0 ad I (7.) s lim P A,M) >_ log - loglog ~ - w = 1 where (7.3) e - a = ee 1- c (i. e. a= c- 1 -log c ad thus a > 0.) Proof of Theorem 7a. We have clearly (7.4) P(dMO >- z) = P f Z k > I < `k>x ad thus by (.18) e x1 (7.5) PO,N() > z) - O - Z 5, k?z M(tk) It follows that if z 1 = 1 log 71-5 a loglog l + co we have (7.6) P(l,N() z zi) = O(e ) This proves (7.1). To prove (7.) we have to estimate the mea ad variace a of r,, where z I = (log - loglog ~ - w. We have by (.18) 1 5 ~` (7.7) M(TZ) eaw - cv7r ad (7.8) D (r,,) = O(M (Zzs». Clearly P(4,N() Z) > P('rz E > 1) = 1 - P(txg = 0) ad it follows from (7.7) ad ('7.8) by the iequality of Chebyshev that (7.9) P( -r x_ = 0) = O(e ). Thus we obtai (7.10) P(4,N() >- z) I - O(e -aw ). Thus (7.) is also proved. N r Remark. If c < 1 the greatest tree which is a compoet of r,n with c is - as metioed above - at the same time the greatest compoet

33 ON THE EVOLUTION OF RANDOM GRAPHS 49 of I' N, as T,N cotais with probability tedig to 1 besides trees oly compoets cotaiig a sigle circle ad beig of moderate size. This follows evidetly from Theorem 4c. As will be see i what follows (see 9) for c > 1 the situatio is completely differet, as i this case r N cotais a very large compoet (i fact of size G(c) with G(c) > 0) which is ot a tree. Note that if we put c = 1 log we have a = 1 log ad 1 log k k k a i coformity- with Theorem c. We ca prove also the followig Theorem 7b. If N- e, where c -/- 1 ad e = ce 1- c the the umber of isolated trees of order h = flog - 5 loglog, + l resp. of order >_ h (where I is a arbitrary real -umber such that h is a positive -iteger) cotaied i F,N has for large it approximately a Poisso distributio with the mea value I = a s 6-al _ a5!' e -al res p. u c Vz cv(i -e - a) does ot cotai a tree of order >_ 1 (log - 5 loglog + l teds to ai a 5/ e -al exp - for. +-, where a = c log c. c ~ /.-,c (1 - e - ) The size of the greatest tree which is a compoet of r N is fairly large if N -. Corollary. The probability that r,n() with N() -c where c This could be guessed from the fact that the costat factor i the expressio 1 log - 5 loglog of the.probable size" of the greatest compoet of F,N figurig i Theorem 7a becomes ifiitely large if c = 1. For the size of the greatest tree i F,N with N - the followig result is valid : Theorem 7c. If N - ad 4,,,N deotes agai the umber of poits of the greatest tree cotaied i I' N, we have for ay sequece co tedig to ~-~ for +-. (7.11) T P(4 N > 1113W) = 0 ad (7.1) lim f r P IdroN > r3 = , 1 4 A Matematikai Kutató Itézet Közleméyei V. A11-.

34 5 0 ERDŐS-RÉNYI Proof of Theorem 7c. We have by some simple computatio usig (7) (7.13) M(z k ) _ Thus it follows that ~ ti ( í kk_ -kl N-k+ f1 ( N = ) = k~~m k k- e -k e 6'. k1 (7.14) P d > -3 w< M t O í1 ( k) - V ~0 which proves (7.11). O the other had, cosiderig the mea ad variace of z* _ it follows that M(a*) z A (03' where A > 0 ad Dz(t*) = 0( 0)3!) ad (7.1) follows by usig agai the iequality of Chebyshev. Thus Theorem 7c is proved. The followig theorem ca be proved by developig further the above argumet ad usig Lemma 1. Theorem 7d. Let r(y) deote the umber of trees of order > y 13 cotaied i r,n() where 0 < y < +- ad N() ti. The we have (7.15) k ~, z" Tk' -,h e -' "M P,N()(T(P) = 9) = - (j = 0, 1,... ) where (7.16) +m = 1 e - x dx. V1 x 3 ; s 8. Whe is F,,,N a plaar graph? We have see that the threshold for a subgraph cotaiig k poits k ad k + d edges is k+a ; thus if N e the probability of the presece of a subgraph havig k poits ad k + d edges i F,N teds to 0 for - +-, for each particular pair of umbers k > 4, d > 1. This however does ot imply that the probability of the presece of a graph of arbitrary order havig more edges tha vertices i r N with N -c teds also to 0 for -{-~. I fact this is ot true for c >_ If as is show by the followig

35 ON THE EVOLUTION OF RANDOM GRAPHS 51 Theorem 8a. Let Y. N (d) deote the umber of cycles of G,N of arbitrary order which are such that exactly d diagoals of the cycle belog also to p,n. The if N() = + ~ + o( V ṉ) where -~ < A < +-, we have e-q (8.1) lim P(Y,N()(d) = y) _ ej (j = 0, 1,... ) -+w j1 where (8.) AY 0 1 a-iel'3 e d y y P=.6a d! o Proof of Theorem 8a. k - go is equal to k(k - 3) We have clearly as the umber of diagoals of a ad thus if N () = + R V + o(v) 4 * k(k-3) l~ ~, N k d, d -- M(x,N(d))=~ 1 ~ l (k - 1)f t JI f k (8.4) M(X,N()(d) }^' a +a ld! d kd-1 I 1 + V e It follows from (8.4) that k=4 l 1 N W a~ ~ 1 ra lim yea-1 e Y3 dy. M(X,N()(d))= 6a d The proof ca be fiished by the same method as used i provig Theorem a. Remark. Note that Theorem 8a implies that if N() = + co V with w + O the the probability that F,,,N() cotais cycles with ay prescribed umber of diagoals teds to 1, while if N() = - w V the same probability teds to 0. This shows agai the fudametal differece i the structure of ',,N betwee the cases N < ad N >. This differ- ece ca be expressed also i the form of the followig Theorem 8h. 0 3 k' Let us suppose that N() -c. If c < 1 the probability

36 5 ERDŐS-RÉNYI that the graph r,n(,) is plaar is tedig to 1 while for c > 1 this probability teds to 0. Proof of Theorem 8b. As well kow trees ad coected graphs cotaiig exactly oe cycle are plaar. Thus the first part of Theorem Sb follows from Theorem 5e. O the other had if a graph cotais a cycle with 3 diagoals such that if these diagoals coect the pairs of poits (P ;, P i ) (i = = 1,, 3) the cyclic order of these poits i the cycle is such that each pair (P ;, P,) dissects the cycle ito two paths which both cotai two of the other poits the the graph is ot plaar. Now it is easy to see that amog the k(k - 3) k. triples of 3 diameters of a give cycle of order k there are at least 6 3 (l triples which have the metioed property ad thus for large values of k approximately oe out of 15 choices of the 3 diagoals will have the metioed property. It follows that if N() _ + CJ', I/r with w +-, the probability that I' N() is ot plaar teds to 1 for -~ +-. This proves Theorem 8b. We ca show that for N() = + 7. with ay real A the probability of r N() ot beig plaar has a positive lower limit, but we caot calculate is value. It may eve be 1, though this seems ulikely. 9. O the growth of the greatest compoet We prove i this (see Theorem 9b) that the size of the greatest compoet of r N() is for N() - e with c > I/ with probability tedig to 1 approximately G(c) where (9.1) G(c) = 1 - x(c) e ad x(c) is defied by (6.4). (The curve y = G(c) is show o Fig. b). Thus by Theorem 6 for N() e with c 1 > / almost all poits of (i. e. all but o() poits) belog either to some small compoet which r N() is a tree (of size at most 1/a (log - loglog) + 0(1) where a = c -1 -log c by Theorem 7a) or to the sigle "giat" compoet of the size -G(C). Thus the situatio ca be summarized as follows : the largest compoet of r N ( ) is of order loge for N() _ c < 1/, of order,'3 1 for () ad of order for INT ( ) N c > 1/. This double "jump" of the size of the largest compoet whe N() passes the value 1/ is oe of the most strikig facts cocerig radom graphs. We prove first the followig

37 0\ THE EVOLUTION OF RANDOM GRAPHS 5 3 Theorem 9a. Let,-r,,,N (A) deote the set of those poits of r,n which belog to compoets of size >A, ad let ff.,n(a) deote the umber of elemets of the set,n (A). If N1 () (c - s) where e > 0, c - s > 1/ ad N () e the with probability tedig to 1 for --->- +- from the H,,N,()(A) poits belogig to.-t,nl()(a) more tha (1 - b) H,N,()(A) poits will be cotaied i the same compoet of ~,N() for ay b with 0 < b < 1 provided that (9.) A > 50 Proof of Theorem 9a. Accordig to Theorem b the umber of poits belogig to trees of order :< A is with probability tedig to 1 for -~ + equal to / q kk-1 E 6 - [(c, - E)]k -1eo (). kl ~.k=1 O the other had, the umber of poits of r N,() belogig to compoets of size _< A ad cotaiig exactly oe cycle is accordig to Theorem 3c o() for c-e > 11 (with probability tedig to 1), while it is easy to see, that the umber of poits of I',Nl() belogig to compoets of size < A ad cotaiig more tha oe cycle is also bouded with probability tedig to 1.) Our last statemet follows by usig the iequality (4.19) from the fact that the average umber of compoets of the metioed type is, as a simple.i calculatio similar to those carried out i previous, shows, of order 0 1 (9.3) Let E1 ) deote the evet that ~H,Nl()(A)-f(A,c-E)I <zf(a,c-e) where r > 0 is a arbitrary small positive umber which will be chose later ad f(a, c) = 1 - c kk- 1 ( ce -C) > 0 ad let E;,1 ) deote the cotrary evet. It follows from what has bee said that (9.5) lim P(E(,1 )) = 0. - ; We cosider oly such r N,(,) for which (9.3) holds. Now clearly I',N,() is obtaied from -P. N I() by addig N()-Y,_() ~--s ew edges at radom to r N,(). The probability that such a ew edge should coect two poits belogig to f H,Nj()(A),Ni()(A), is at least I1 (~ - AT() ad thus by (9.3) is ot less tha (1 - z) f (A, c - e), if is sufficietly large ad 'r sufficietly small.

38 5 4 ERDŐs -Rr:NYI As these edges are chose idepedetly from each other, it follows by the law of large umbers that deotig by v the umber of those of the N () - N1 () ew edges which coect two poits of.nl() ad by E() the evet that (9.6) v >_ E(1-3r) J (A, c - E) ad by E) the cotrary evet, we have (9.7) lim P(E)) = We cosider ow oly such r N() for which E) takes place. -Now let us cosider the subgraph r NE() of I',N() formed by the poits of the set,n (,,)(A) ad oly of those edges of r. N,(,) which coect two such poits. 'We shall eed ow the followig elemetary r Lemma. Let a l, a,..., ar be positive umbers, Z aj = 1. If max a, :< a the there ca be foud a value k (1 < k s r -- 1) such that (9.8) ad 1- a k 1 -{- a _ <- j= ] a~ ` 1-a < 1+a - )' aj <_ j=k+1 j=1 1<j<r Proof of Lemma. Put Sj = a (j = 1,,..., r). Let jo deote the least iteger, for which Sj > 1/. I case Sjo - 1/ > 1 / - Sjo_ 1 choose k = jo - 1, while i case Sao ḻ 1 / S 1 / - S1o _ 1 choose k = jo. I both cases we have I Sk - 1/ 1 < amp < a which proves our Lemma. Let the sizes of the compoets of F N() be deoted by b1, b,..., br. Let E3) deote the evet (9.9) max bj > H,NI()(A) (1 - S) ad E(,,3) the cotrary evet. Applyig our Lemma with a = 1 - b to the umbers a j = bj it follows that if the evet E (3) takes place, the H,Nl()(A) Set ` N,()(A) ca be split i two subsets A ad r cotaiig H ad H poits such that H } H = H, NI()(A) ad - (9.10) H,,Nl()(A) S mi (H, H"t ) < max (H, H) < H,,,N,()(A) I I -, further o poit of. ' is coected with a poit of r i F*,N() It follows that if a poit P of the set '-- Nl() (A) belogs to (resp. te} the all other poits of the compoet of I',NI() to which P belogs are

39 ON THE EVOLUTION OF RANDOM GRAPHS 5 5 also cotaied i r (resp. rf). As the umber of compoets of size > A Of r,ni() is clearly < H,N )(A) th e umber of such divisios of the set 1 xr ~. ~>(A) does ot exceed A -,N,()(A) If further E3 ) takes place the every oe of the v ew edges coectig poits of.n,()(a) coects either two poits of or two poits of fi. The possible umber of such choices of these edges is clearly ( l + I rz ) As by (9.10) H, w ~H 1,1< b.- + (11á+b<1a =-- 4 l 1 it follows that (9.1) ad thus by (9.3) ad (9.6) 1 1 H,.v,17(A) á P(E(3 ) < A ( 1 - ~e(1-3r)f(a,e-e) (9.13) P(E3» < exp 1f(A, c - e) ~(1 +-r) log e(1-3 i) f(á, e s) á A 1 Thus if (9.14) A sá(1-3 z) f(a, c - s) > (1 + z) log o the (9.15) 1im P(EW) = 0. om+ As however i case c - e > i/ we have I(A, c - e) > G (c - s) > 0 for ay A, while i case c - s -. 1 / kk-1 kk-1 >_ (9.15x) f 1 =I- = 1 la,. k=1 k 1 Ck k= A+, k! ek VA if A z A o the iequality (9.13) will be satisfied provided that r < 1 ad A > Eá Thus Theorem 9a is proved. Clearly the "giat" compoet of r,n() the existece of which (with probability tedig to 1) has bee ow proved, cotais more tha (1 -T)(1 -á)f (A,c-s)

40 5 6 ERDŐS-E (;N l I poits. By choosig e, t ad 6 sufficietly small ad A sufficietly large, (1 - z) (1 - S) f(a, c - E) ca be brought as ear to G(c) as we wat. Thus we have icidetally proved also the followig Theorem 9b. Let e,,,, deote the size of the greatest compoet of r,n' If V() - e where c > lf we have for ay 7) > 0 (9.16) lim P Q,N() -G(c) -- ~,' k 1 where G(c) = 1 - x(c) ad x(c) _ - (c e-c)k is the solutio satisfyig c k-1 k. 0 < x(c) < 1 of the equatio x(c) e - Y(c) = ce -c Remark. As G(c) --->- 1 for c -* + - it follows as a corollary- from Theorem 9b that the size of the largest compoet will exceed (1 -a) if c is sufficietly large where a > 0 is arbitrarily small. This of course could be proved directly. As a matter of fact, if the greatest compoet of r,,,n(,,) with N() -ve would ot exceed (1 - a) (we - deote this evet by B (a, c)) oe could by Lemma divide the set V of the poits Pl,..., P i two subsets P resp. V" cosistig of ' resp " - poits so that o two poits belogig to differet subsets are coected ad a (9.17) < mi (', ') < max W, W) < I ~ But the umber of such divisios does ot exceed 1, ad if the poits are divided i this way, the umber of ways N edges ca be chose so that oly poits belogig to the same subset V' resp. V" are coected, is '~ + ~"~) As (.'` + ("` < a ~ _ I 9 ~ ~. it follows l JI L Cl N () N()a (9.18) P(B,(a, c)) < ' ~1 - -I < 1 e ; Thus if a e > log4, the (9.19) lai P(B,(a, c)) = 0 which implies that for c > log 4 ad AT() c we have a (9.0) lim P(O,N() >_ (1-60) = 1.

41 ON THE EVOLUTION OF RANDOM GRAPHS 57 'e have see that for N() - e with c > 11 the radom graph r,n() cosists with probability tedig to 1, eglectig l o() poits, oly of isolated trees (there beig approximately k k (c e-c)k c k! trees of order k) ad of a sigle giat compoet of size G(c). Clearly the isolated trees melt oe after aother ito the giat compoet, the "dager" of beig absorbed by the "giat" beig greater for larger compoets. As show by Theorem e for N() 1 log oly isolated k trees of order < k survive, while for N() - 1/ log - ~ +- the whole graph will with probability tedig to 1 be coected. A iterestig questio is : what is the "life-time" distributio of a isolated tree of order k which is preset for AT() -c? This questio is aswered by the followig Theorem 9r,. The probability that a isolated tree of order k which is preset i. 1,,N,(.) where N,() -~- c ad c > 1/ should still remai a isolated tree i r M() where í1' () (c + t) (t > 0) is approximately e -kt ; thus the Life-time" of a tree of order k has approximately a expoetial distributio with mea value ad is idepedet of the "age" of the tree. k - Proof. The probability that o poit of the tree i questio will be coected with ay other poit is This proves Theorem 9e. (-k` N,() I k, e 1, -kt l=n,()+i ~~ Remarks ad some usolved problems We studied i detail the evolutio of " N oly till N reaches the order of magitude log. (Oly Theorem 1 embraces some problems cocerig the rage N() -, with 1 < a <.) We wat to deal with the structure of rn() for N() - c with a > 1 i greater detail i a fortcomig paper ; here we make i this directio oly a few remarks. First it is easy to see that -P,,(r) N() is really othig else, tha the complemetary graph of F,N() Thus each of our results ca be reformulated to give a result o the probable structure of ",,N with N beig ot much less tha I For istace, the structure of r N will have a secod abrupt chage whe ű' passes the value ; if N < ~l -e with C>1/ the the complemetary graph of -F,,N will cotai a coected graph of order f(c), while for c < i/ this (missig) "giat" will disappear.

42 5 8 ERDŐS-RÉNYI To show a less obvious example of this priciple of gettig result for N ear to ~, let us cosider the maximal umber of pairwise idepedet poits i r,n (The vertices P ad Q of the graph F are called idepedet if they are ot coected by a edge). Evidetly if a set of k poits is idepedet i r,n() the the same poits form a complete subgraph i the complemetary graph r,n() As however r,n() has the same structure as F (Z) _ N() it follows by Theorem 1, that there will be i r,n() almost surely o k idepedet poits if r, - k -- N() = o I(1 11) il e. if N() _ - k o (1-1i.) but there will be i r, i r,n() almost surely k idepedet poits if N() _ (~ - co,, 0 - k-i- where co teds arbitrarily slowly to +-. A other iterestig questio is : what ca be said about the degrees of the vertices of r,n We prove i this directio the followig Theorem 10. Let D,N()(Pk) deote the degree of the poit P k i r,n() (i. e. the umber of poits of r,n() which are coected with P k by a edge). Put Suppose that D = mi D,,,N() (P k ) ad D = max D,N() (Pk) I<k :s~ t<ks (10.1) lim N() = T o0 + m log The we have for ay e > o (10.) lim e f D We have further for N() ~-ca for ay k (10.3) +~ -1 D lim p ( c)j e-c P(D,N() ( k) _%) _ ~ (i - 0, 1,... ) --+ -? Proof. The probability that a give vertex P k shall be coected by exactly r others i r,n is -1-1) h~n~r - N e i r N-r, N. (~ r!, N

43 OA THE EVOLUTION OF RANDOM GRAPHS 59 thus if N() -o the degree of a give poit has approximately a Poisso distributio with mea value c. The umber of poits havig the degree r is thus i this case approximately ( C)r e -c - (r = 0,1,.. ) r? If N() _ ( log ) w with co -* +- the the probability that the degree of a poit will be outside the iterval N() (1 - e) ad N() (1 + E) is ap- proximately ( o) log' ke - w log -O ( 1,k-1og.w,,>e.log w,i k. eywj ad thus this probability is o 1, for ay E > 0. [ 1 Thus the probability that the degrees of ot all poits will be betwee the limit (1 ± E) w log will be tedig to 0. Thus the assertio of Theorem 10 follows. A iterestig questio is : what will be the chromatic umber of r,ly '? (The chromatic umber Ch(P) of a graph r is the least positive iteger h such that the vertices of the graph ca be coloured by h colours so that o two vertices which are coected by a edge should have the same colour.) Clearly every tree ca be coloured by colours, ad thus by Theorem 4a almost surely Ch (.T N) - if N = o(). As however the chromatic umber of a graph havig a equal umber of vertices ad edges is equal to or 3 accordig to whether the oly cycle cotaied i such a graph is of eve or odd order, it follows from Theorem 5e that almost surely Ch (I' p,) < 3 for N() - o with c < IJa For N() ti we have almost surely Ch (P tvc>) > 3. As a matter of fact, i the same way, as we proved Theorem 5b, oe ca prove that r N() cotais for N() - almost surely a cycle of odd order. It is a ope problem how large Ch (P N()) is for N(a) -o withc> 1/? A further result o the chromatic umber ca be deduced from our above remark o idepedet vertices. If a graph T has the chromatic umber h, the its poits ca be divided ito h classes, so that o two poits of the same class are coected by a edge ; as the largest class has at least poits, h it follows that if f is the maximal umber of idepedet vertices of r we have f >_. IVOw we have see that for N() = _+ (1 k) I almost surely h J j f < k ; it follows that for N() = t.(1 - o ( k~l almost surely Ch (r,ly()) >

44 6 0 ERDŐS-RÉNYI Other ope problems are the followig : for what order of magitude of N() has r,~~ 1 with probability tedig to 1 a Hamilto-lie (i.e. a path which passes through all vertices) resp. i case is eve a factor of degree 1 (i.e. a set of disjoit edges which cotai all vertices). A other iterestig questio is : what is the threshold for the appearace of a "topological complete graph of order k" i.e. of k poits such that ay two of them ca be coected by a path ad these paths do ot itersect. For k > 4 we do ot kow the solutio of this questio. For k = 4 it follows from Theorem 8a that the threshold is. It is iterestig to compare this with a (upublished) result of G. DIRAC accordig to which if N >_ : - the GT,N cotais certaily a topological complete graph of order 4. We hope to retur to the above metioed usolved questios i a other paper. Remark added o May 16, It should be metioed that N. V. SMIRNOV (see e. g. Mare~uamugecmwi C6opuux 6(1939) p. 6) has proved a lemma which is similar to our Lemma 1. (Received December 8, 1959.) REFERENCES CaYLEY, A. : Collected Mathematical Papers. Cambridge, PóLYa, G. : "Kombiatorische Azahlbestimmuge für Gruppe, Graphe ad chemisebe Vcrbiduge". Acta Mathematics 68 (H37) KÖNIG, D.: Theorie der edliche ati uedliche Graphe. Leipzig, BERGE, C. : Théorie des graphes et ses applicatios. Paris, Duod, FORD, G. W.-UHLENBECK, G. E. : "Combiatorial problems i the theory of graphs, I." Proc. Nat. Acad. Sci. 4 (1956) USA GILBERT, E. N. : "Eumeratio of labelled graphs". Caadia Joural o f Math. 8 (1957) ERDős, P.-RÉNYI, A. : "O radom graphs, I". Publicatioes Mathematicae (Debrece) 6 (1959) HARaRY, F. : "Usolved problems i the eumeratio of graphs" I this issue, p. 63. ERDős, P. : Graph theory ad probability." Caadia Joural of Math. 11 (1959) AUSTIN, T. L.-FAGEN, R. E.-PENNEY, `V. F.-RIORDAN, J. : "The umber of compoets i radom liear graphs". Aals of Math. Statistics 30 (1959) TURÁN P.: "Egy gráfelméleti szélsőérték feladatról". Matematikai és Fizikai Lapok 48 (1941) TURÁN, P. : "O the theory of graphs", Colloquium Mathematicum 3 (1954) KŐVÁRY, T.-Sós, V. T.-TURÁN. P.: "O a problem of K. Zarekiewicz". Colloquium Mathematicum 3 (1954) RÉNYI, A. : "Some remarks o the theory of trees". Publicatios of the Math. Ist. of the Hug. Acad. of Sci. 4 (1959) RÉNYI, A. : "O coected graphs, I.".Publicatios of the Math. Ist. of the Hug. Acad. of Sci. 4 (1959) JORDAN, CH. : Calculus of fiite differeces. Budapest, ERDŐS, P.-STONE, A. H. : "O the structure of liear graphs". Bull. Amer. Math. Soc. 5 (1946)

45 61 0 РАЗВЁРТы ВАНИЕ СЛУцАЙНы Х ГРАФОВ P. ERDŐS и А. RÉNYI Резюме Пусть дaны точки Р 1, Р,..., Р, и выбираем случайно друг за другом N из возможны х ~ ребер (Р~, Р1 ) тaк что после того что вы брани k ребра каждый из других - k ребер имеет одинаковую вероятность быть выlп бранным как следующий. Работа занимается вероятной структурой так получаемого слуцайного графa Г, N при условии, что N = N(п) известнaя функция от и очень большое число. Особенно исслeдуется изьченение этой структуры если N нарастает при данном очень большом. Случайно развёртывающий граф может быть pассмотрен как упрощенный модель pостa реальны x сетей (нaпример сетей связи).

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