Lecture 2 Long paths in random graphs

Transcription

1 Lecture Log paths i radom graphs 1 Itroductio I this lecture we treat the appearace of log paths ad cycles i sparse radom graphs. will wor with the probability space G(, p) of biomial radom graphs, aalogous results for the sister model G(, m) ca be either prove usig very similar argumets, or derived usig available equivalece statemets betwee the two models. Our goal here is two-fold: we first prove that already i the super-critical regime p = 1+ɛ, the radom graph G(, p) cotais typically a path of legth liear i ; the we prove that i the regime p = C, the radom graph G(, p) has typically a path, coverig the proportio of vertices tedig to 1 as the costat C icreases. We will ivoe the approaches ad results developed i Lecture 1 (the Depth First Search Algorithm ad its cosequeces) to achieve both of these goals, ad i fact i a rather short ad elegat way. We Liearly log paths i the supercritical regime I their groudbreaig paper [] from 1960, Paul Erdős ad Alfréd Réyi made the followig fudametal discovery: the radom graph G(, p) udergoes a remarable phase trasitio aroud the edge probability p() = 1 1 ɛ. For ay costat ɛ > 0, if p =, the G(, p) has whp all coected compoets of size at most logarithmic i, while for p = 1+ɛ whp a coected compoet of liear size, usually called the giat compoet, emerges i G(, p) (they also showed that whp there is a uique liear sized compoet). The Erdős-Réyi paper, which lauched the moder theory of radom graphs, has had eormous ifluece o the developmet of the field. We will be able to derive both parts of this result very soo. Although for the super-critical case p = 1+ɛ the result of Erdős ad Réyi shows a typical existece of a liear sized coected compoet, it does ow imply that a logest path i such a radom graph is whp liearly log. This was established some 0 years later by Ajtai, Komlós ad Szemerédi [1]. I this sectio we preset a (relatively) simple proof of their result. We will ot attempt to achieve the best possible absolute costats, aimig rather for simplicity. Our treatmet follows closely that of []. 1

2 The most fudametal idea of the proof is to ru the DFS algorithm o a radom graph G G(, p), costructig the graph o the fly, as the algorithm progresses. We first fix the order σ o V (G) = [] to be the idetity permutatio. Whe the DFS algorithm is fed with a sequece of i.i.d. Beroulli(p) radom variables X = (X i ) N i=1, so that is gets its i-th query aswered positively if X i = 1 ad aswered egatively otherwise, the so obtaied graph is clearly distributed accordig to G(, p). Thus, studyig the compoet structure of G ca be reduced to studyig the properties of the radom sequece X. This is a very useful tric, as it allows to flatte the radom graph by replacig a iheretly two-dimesioal structure (a graph) by a oe-dimesioal oe (a sequece of bits). I particular, observe crucially that as log as T, every positive aswer to a query results i a vertex beig moved from T to U, ad thus after t queries ad assumig T still, we have S U t i=1 X i. (The last iequality is strict i fact as the first vertex of each coected compoet is moved from T to U for free, i.e., without eed to get a positive aswer to a query.) O the other had, sice the additio of every vertex, but the first oe i a coected compoet, to U is caused by a positive aswer to a query, we have at time t: U 1 + t i=1 X i. The probabilistic part of our argumet is provided by the followig quite simple lemma. Lemma.1 Let ɛ > 0 be a small eough costat. Cosider the sequece X = (X i ) N i=1 Beroulli radom variables with parameter p. of i.i.d. 1. Let p = 1 ɛ. Let = 7 l. The whp there is o iterval of legth i [N], i which at ɛ least of the radom variables X i tae value 1.. Let p = 1+ɛ. Let N 0 = ɛ. The whp N0 i=1 X i ɛ(1+ɛ) /. Proof 1) For a give iterval I of legth i [N], the sum i I X i is distributed biomially with parameters ad p. Applyig the Cheroff boud to the upper tail of B(, p), ad the the uio boud, we see that the probability of the existece of a iterval violatig the assertio of the lemma is at most for small eough ɛ > 0. (N + 1)P r[b(, p) ] < e ɛ (1 ɛ) < e ɛ (1 ɛ) 7 ɛ l = o(1), ) The sum N 0 i=1 X i is distributed biomially with parameters N 0 ad p. Hece, its expectatio is N 0 p = ɛ p = ɛ(1+ɛ), ad its stadard deviatio is of order. Applyig the Chebyshev iequality, we get the required estimate. Now we are ready to formulate ad to prove the result of this sectio. Theorem. Let ɛ > 0 be a small eough costat. Let G G(, p). 1. Let p = 1 ɛ. The whp all coected compoets of G are of size at most 7 ɛ l.. Let p = 1+ɛ. The whp G cotais a path of legth at least ɛ.

3 I both cases, we ru the DFS algorithm o G G(, p), ad assume that the sequece X = (X i ) N i=1 of radom variables, defiig the radom graph G G(, p) ad guidig the DFS algorithm, satisfies the correspodig part of Lemma.1. Proof 1) Assume to the cotrary that G cotais a coected compoet C with more tha = 7 ɛ l vertices. Let us loo at the epoch of the DFS whe C was created. Cosider the momet iside this epoch whe the algorithm has foud the ( + 1)-st vertex of C ad is about to move it to U. Deote S = S C at that momet. The S U =, ad thus the algorithm got exactly positive aswers to its queries to radom variables X i durig the epoch, with each positive aswer beig resposible for revealig a ew vertex of C, after the first vertex of C was put ito U i the begiig of the epoch. At that momet durig the epoch oly pairs of edges touchig S U have bee queried, ad the umber of such pairs is therefore at most ( ) + ( ) <. It thus follows that the sequece X cotais a iterval of legth at most with at least 1 s iside a cotradictio to Property 1 of Lemma.1. ) Assume that the sequece X satisfies Property of Lemma.1. We claim that after the first N 0 = ɛ queries of the DFS algorithm, the set U cotais at least ɛ vertices (with the cotets of U formig a path of desired legth at that momet). Observe first that S < at time N 0. Ideed, if S, the let us loo at a momet t where S = (such a momet surely exists as vertices flow to S oe by oe). At that momet U 1 + t i=1 X i < by Property of Lemma.1. The T = S U, ad the algorithm has examied all S T 9 > N 0 pairs betwee S ad T (ad foud them to be o-edges) a cotradictio. Let us retur to time N 0. If S < ad U < ɛ the, we have T. This meas i particular that the algorithm is still revealig the coected compoets of G, ad each positive aswer it got resulted i movig a vertex from T to U (some of these vertices may have already moved further from U to S). By Property of Lemma.1 the umber of positive aswers at that poit is at least ɛ(1+ɛ) /. ( Hece we have ) S U ɛ(1+ɛ) /. If U ɛ, the S ɛ + ɛ 10 /. All S T S S ɛ pairs betwee S ad T have bee probed by the algorithm (ad aswered i the egative). We thus get: ɛ ( ) ( ) ( = N 0 S S ɛ ɛ + ɛ 10 / ɛ ) ɛ + / = ɛ + ɛ 0 O(ɛ ) > ɛ (we used the assumptio S < ), ad this is obviously a cotradictio, completig the proof. Let us discuss the obtaied result ad its proof. First, give the probable existece of a log path i G(, p), that of a log cycle is just oe short step further. Ideed, we ca use sprilig as follows. Let p = 1+ɛ for small ɛ > 0. Write 1 p = (1 p 1)(1 p ) with p = ɛ ; thus, most of. Let ow G G(, p), G 1 G(, p 1 ), G G(, p ), we ca the probability p goes ito p 1 1+ɛ/ represet G = G 1 G. By Theorem., G 1 whp cotais a liearly log path P. Now, the edges

4 of G ca be used to close most of P ito a cycle there is whp a edge of G betwee the first ad the last / (say) vertices of P. The depedecies o ɛ i both parts of Theorem. are of the correct order of magitude for p = 1 ɛ a largest coected compoet of G(, p) is ow to be whp of size Θ(ɛ ) log while for p = 1+ɛ a logest cycle of G(, p) is whp of legth Θ(ɛ ). Observe that usig a Cheroff-type boud for the tales of the biomial radom variable istead of the Chebyshev iequality would allow to claim i the secod part of Lemma.1 that the sum N0 i=1 X i is close to ɛ(1+ɛ) with probability expoetially close to 1. This would show i tur, employig the argumet of Theorem., that G(, p) with p = 1+ɛ cotais a path of legth liear i with expoetially high probability, amely, with probability 1 exp{ c(ɛ)}. As we have metioed i Lecture 1, the DFS algorithm is applicable equally well to directed graphs. Hece essetially the same argumet as above, with obvious mior chages, ca be applied to the model D(, p) of radom digraphs. It the yields the followig theorem: Theorem. Let p = 1+ɛ, for ɛ > 0 costat. directed path ad a directed cycle of legth Θ(ɛ ). The the radom digraph D(, p) has whp a This recovers the result of Karp [4]. Nearly spaig paths Cosider ow the regime p = C, where C is a (large) costat. Our goal is to prove that whp i G(, p), the legth of a logest path approaches as C teds to ifiity. This too is a classical result due to Ajtai, Komlós ad Szemerédi [1], ad idepedetly due to Feradez de la Vega []. It is fairly amusig to see how easily it ca be derived usig the DFS-based tools we developed i Lecture 1. Theorem.1 For every ɛ > 0 be a small eough costat there exists C = C(ɛ) > 0 such that the followig is true. Let G G(, p), where p = C. The whp G cotais a path of legth at least (1 ɛ). Proof Let = ɛ. By Propositio. from Lecture 1 it suffices to show that G G(, p) cotais whp a edge betwee every pair of disjoit subsets of size of V (G). For a give pair of disjoit sets S, T of size S = T =, the probability that G cotais o edges betwee S ad T is exactly (1 p) (all pairs betwee S ad T come out o-edges i G). Usig the uio boud, we obtai that the probability of the existece of a pair violatig this requiremet is at most ( )( ) ( ) ( e ) [ (e ) ] (1 p) < (1 p) < e p( 1) < e C( 1). Recallig the value of ad taig C = l(/ɛ ) ɛ guaratees that the above estimate vaishes (i fact expoetially fast i ), thus establishig the claim. 4

5 A similar statemet holds for the probability space D(, p) of radom directed graphs, with a essetially idetical proof. Refereces [1] M. Ajtai, J. Komlós ad E. Szemerédi, The logest path i a radom graph, Combiatorica 1 (1981), 1 1. [] P. Erdős ad A. Réyi, O the evolutio of radom graphs, Publ. Math. Ist. Hugar. Acad. Sci. (1960), [] W. Feradez de la Vega, Log paths i radom graphs, Studia Sci. Math. Hugar. 14 (1979), 40. [4] R. Karp, The trasitive closure of a radom digraph, Radom Structures ad Algorithms 1 (1990), 7 9. [] M. Krivelevich ad B. Sudaov, The phase trasitio i radom graphs a simple proof, Radom Structures ad Algorithms, to appear.