S.Janson, T./Luczak and A.Ruci'nski, An exponential bound for the probability of nonexistence of a specified subgraph of a random graph Proceedings of Random Graphs' 87 Wiley, Chichester 1990, 73-87.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....i] e i =i ; i = 0; 1; P r[P (1 ) e =2 (3) see, e.g. Theorem A.15 of [2] When X is the sum of many mostly independent indicator random variables, there is some hope that X is also highly concentrated around its mean as a binomial random variable. Recent results ([9], 7] 8] 12] give su cient conditions for this phenomenon to hold. Below we describe a general framework for applying these results. Suppose Q is a nite set ( in our instances Q is the edge set of a complete graph on n vertices) Let fJ i : i 2 Qg be a set of independent random indicator ....

....1) 2 ) 10) where c and c are constants depending only on the values of r and s and in the particular case r = 3; s = 4 f 3;4 (n) cn 2=3 1=3 ; 11) where c is an absolute constant. The bounds (10) and (11) were obtained by combining the local lemma with the Janson inequality([9]) Some constructive bounds for f r;s (n) are presented in [1] Here we improve the bound (11) using the technique based on large deviation inequalities. Our treatment is rather similar to that of previous section and can be used to improve (10) in a similar manner. 10 Theorem 2 There exists an ....

S. Janson, T. Luczak and A. Rucinski, An exponential bound for the probability of nonexistence of a speci ed subgraph in a random graph, in Random Graphs' 87 (M. Karonski et al., eds.), Wiley, Chichester,

....p) has property (#, a) Lemma 2 allows us to require that (#, a) should hold for all G # G. Now we are ready to establish the existence of T = T (K) for all K # K. Indeed, for any fixed K the fact that Base(K) contains at least a V (G) triangles implies, via Janson s inequality [11], that, with probability 1 , there is a triangle T (K) in Base(K) G(n, #p) Since (i) holds, we have that a suitable triangle T = T (K) does exist a.a.s. for every K # K. The conclusion of (b) now follows from (iii ) To summarize, the convergence follows indirectly from a ....

Svante Janson, Tomasz # Luczak, and Andrzej Rucinski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, Random graphs '87 (Poznan, 1987.

....1 Exponential Haystacks 1.1 Janson Inequalities Let A 1 ; Am be events in a probability space. Set m Y i=1 Pr[A i ] The Janson Inequality allows us, sometimes, to estimate Pr[A i ] by M , the probability if the A i were mutually independent. The original proof by Svante Janson is in [13]. See [5] for a more elementary proof and [2] for general discussion. We let G be a dependency graph for the events i.e. the vertices are the indices i 2 [m] and each A i is mutually independent of all A j with j not adjacent to i in G. This notion was first used with the Lov asz Local ....

....edges whatsoever and so, for n p 1 Pr[TF ] 1 Gamma p) With a bit more care, in fact, one can estimate Pr[TF ] up to a constant in the logarithm for all p. These methods do not work just for trianglefreeness. In a remarkable paper Andrzej Rucinski, Tomasz Luczak and Svante Janson [13] have examined the probability that G(n; p) does not contain a copy of H , where H is any particular fixed graph, and they estimate this probability, up to a constant in the logarithm, for the entire range of p. Their paper was the first and still one of the most exciting applications of the ....

S. Janson, T. / Luczak and A. Rucinski. An exponential bound for the probability of nonexistence of specified subgraphs of a random graph, in Procedings of Random Graphs '87, M. Karonski et. al. eds, J. Wiley 1990, 73-87

....X, though with particular emphasis at X = 0. For example, when p = c=n and A ijk = ffi; jg; fi; kg; fj; kgg range over all triangles these conditions hold and G(n; p) is trianglefree with probability exp( Gammac =6) as known to Erdos and R enyi. Sweeping generalizations of this are given in [13] where the first proof of Janson s Inequality may be found. Other proofs and generalizations are given in [14] 2] Applying Janson to Pr[ G(n; 5) k] we let A S = S] S ranging over the k sets of vertices. Then ffl 0, f(k) Delta is the expected number of edge overlapping ....

S. Janson, T. / Luczak and A. Rucinski. An exponential bound for the probability of nonexistence of specified subgraphs of a random graph, in Procedings of Random Graphs '87, M. Karonski et. al. eds, J. Wiley 1990, 73-87

....t) log(n=t) tf1 o(1)g, with the most significant improvement being when t = n . The versatility of Janson s inequalities in combinatorial situations has been welldocumented; see, for example, the wide range of examples in Chapter 8 of [1] or the work of Janson, Luczak, and Ruci nski [12], who establish the definitive threshold results for Tur an type properties in the unipartite case. Recent applications of these exponential inequalities include an an analysis of the threshold behaviour of random covering designs ( 10] of random Sidon sequences ( 14] and of the Schur property ....

S. JANSON, T. / LUCZAK, AND A. RUCI ' NSKI, An exponential bound for the probability of non-existence of a specified subgraph in a random graph, in " Proceedings of 13 the 1987.

....that there is an (s i ; t i ) path for each i. Thus we see an application of the framework of Theorem 4.3 to a family of hard lowcongestion routing problems. 31 5 Analyses via the Janson Luczak Ruci nski inequality A major breakthrough concerning random graphs was made in the work of [34]. A key point from there of relevance in our context is the handling of results in a direction opposite to that of the FKG inequality. For instance, for certain types of decreasing events E 1 ; E 2 ; E k , the probability of all the E i occurring is upper bounded well in [34] a reasonable ....

....in the work of [34] A key point from there of relevance in our context is the handling of results in a direction opposite to that of the FKG inequality. For instance, for certain types of decreasing events E 1 ; E 2 ; E k , the probability of all the E i occurring is upper bounded well in [34] (a reasonable lower bound is immediate via FKG) We refer the reader to [34] and to Chapter 8 of [4] for more details. A result related to these has been given a very simple and elegant proof by Boppana Spencer [16] their proof approach is one of the main motivations for some of the work of ....

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S. Janson, T. / Luczak and A. Ruci'nski. An exponential bound for the probability of nonexistence of a specified subgraph in a random graph. In Random Graphs '87 (M. Karo'nski, J. Jaworski and A. Ruci'nski, eds.), John Wiley & Sons, Chichester, pages 73--87, 1990.

.... of such graphs is n bn=2c bn 2 =4c m : 3 For m = o(n 4=3 ) the best bounds on jT (n; m)j are due to Wormald [20] If m=n 4=3 6 0 and if m is not large enough for Theorem 2 to apply, the best bounds are those which follow from the results in Janson, uczak and Ruciski [6]. Finally, it turns out that the proof of Theorem 3 can be extended to work for any odd cycle. Theorem 5. Given an odd integer , let t = t (n) 1 n 2 log n 1= 1) Then for any 0, P [G n;m is bipartite] P [G n;m is C free] 8 : 1 if m = o(n) 0 if ....

....1 statement of Theorem 3. We stress that most of the notions and statements are only made precise in the later sections. In contrast to [16] and [12] the main tools used in our proof of the second 1 statement are the correlation inequalities (31) and (32) which are from Janson, uczak and Ruciski [6]. In [6] these were applied to give an exponential upper bound on the probability that a random graph G n;p (and thus G n;m ) is H free for some xed graph H, where G n;p denotes a random graph with n vertices and edge probability p. For m t 3 and H a triangle, the bound obtained in this way ....

[Article contains additional citation context not shown here]

S. Janson, T. uczak and A. Ruciski, An exponential bound for the probability of nonexistence of a specied subgraph in a random graph, in Random Graphs '87, eds. M. Karoski, J. Jaworski and A. Ruciski, John Wiley & Sons (1987), 7387.

....P r[X (1 )np] e 2 np=2 ; 3) P r[X (1 )np] e 2 (1 )np=2 : 4) When X is the sum of many rarely dependent indicator random variables, it is also possible in certain cases to obtain exponential bounds on the tails of X. Let us describe a general scheme rst presented in [6]. Suppose Q is a nite universal set (in our instances Q is the edge set of a complete graph on N vertices) Let fJ i : i 2 Qg be a set of independent indicator random variables, P r[J i = 1] p i for every i 2 Q (J i = 1 if the corresponding edge belongs to E(G) where G is a random graph on N ....

S. Janson, T. Luczak and A. Rucinski, An exponential bound for the probability of nonexistence of a specied subgraph in a random graph, in Random Graphs' 87 (M. Karonski et al., eds.), Wiley, Chichester, 1990, 73-87.

....of the presented approach for estimating the threshold for the existence of an H factor for other graphs H, in particular, for the case H = K r , r 3. 2 The Janson inequality In the course of the proof we will make a multiple use of the powerful inequality of Janson, rst described in [4] (see also [1] Ch. 8) The following particular scheme of the inequality will suce for our purposes. Let S be a family of labelled subgraphs of a complete graph on n labelled vertices. Each edge of this complete graph is chosen to be an edge of a random graph G 2 G(n; p) with probability p, all ....

S. Janson, T. Luczak and A. Rucinski, An exponential bound for the probability of nonexistence of a specied subgraph in a random graph, in Random Graphs' 87 (M. Karonski et al., eds.), Wiley, Chichester, 1990, 73-87.

....corollary of our results is that for families of instances of both PIPs and CIPs, we get a good (O(1) or 1 o(1) integrality gap, if B grows at least as fast as log a. For PIPs and CIPs, we use the FKG inequality in addition to Theorem 3. 1; for PIPs, we also need the powerful Janson s inequality ([16, 9], see also Chapter 8 of [4] Bounds on the result of a greedy algorithm for CIPs relative to the optimal integral solution, are known [11, 12] Our bound improves that of [11] and is incomparable with [12] for any given A, c, and the unit vector b=jjbjj 2 , our bound improves on [12] if B is ....

....integer programs. For PIPs and CIPs, we extend some of the ideas of Theorem 3.1, and also use some correlation results of [31] A crucial way in which we will need to extend Theorem 3. 1 for PIPs is in allowing one E i , Em 1 , to have C i;j s that can become negative: we use Janson s inequality [16, 9] to aid in this. For PIPs, we solve the LP relaxation and set x 0 i : x i =ff for some ff 1 to be fixed later; this scaling down is done to boost the chance that the constraints in the PIP are all satisfied. Define a random z 2 Z m , the outcome of randomized rounding, as follows. ....

S. Janson, T. / Luczak and A. Rucinski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, In Random Graphs '87, Karonski et al., eds., John Wiley & Sons, Chichester, 1990, pp. 73--87.

....R 2 R(n; p) that are typical and non exceptional. Thus H = H(R) will always consist of isolated edges. To bound the contribution of the typical, non exceptional R 0 [n] to the sum in (6) we shall need the following consequence of a large deviations inequality of Janson, Luczak, and Ruci nski [J LR 90] Let J be a graph with maximal degree at most 2, with m edges, and n 2 vertices of degree two. Let J p be a random induced subgraph of J obtained by selecting its vertices randomly and independently each with probability p. Then the inequality of Janson, Luczak and Ruci nski gives that the ....

Janson, S., Luczak, T. and Rucinski, A., An exponential bound for the probability of nonexistence of a specied subgraph in a random graph, in \Random Graphs '87," (Karonski, M., Jaworski, J. and Rucinski, A., eds), Wiley, 1990, pp. 73-87.

....improves on the bounds which follow from Theorem 2 for these m. For m = o(n 4=3 ) the best bounds on jT (n; m)j are due to Wormald [16] If n 4=3 = O(m) and if m is not large enough for Theorem 2 to apply, the best bounds are those which follow from the results in Janson, uczak and Ruciski [5]. Theorem 4. Almost all bipartite graphs with n vertices and m 20n log n edges have a bipartition where the size of the vertex classes di ers by at most n log n= p m. Moreover, if in addition m = o(n 2 ) the number of such graphs is (1 o(1) p 4 n p m n bn=2c bn 2 ....

....1 statement of Theorem 3. We stress that most of the notions and statements are only made precise in the later sections. In contrast to [13] and [10] the main tool used in our proof of the second 1 statement are the correlation inequalities (25) and (26) which are from Janson, uczak and Ruciski [5]. In [5] these were applied to give an exponential upper bound on the probability that a random graph G n;p (and thus G n;m ) is H free for some xed graph H, where G n;p denotes a random graph with n vertices and edge probability p. For m t 3 and H a triangle, the bound obtained in this way is ....

[Article contains additional citation context not shown here]

S. Janson, T. uczak and A. Ruciski, An exponential bound for the probability of nonexistence of a specied subgraph in a random graph, in Random Graphs '87, eds. M. Karoski, J. Jaworski and A. Ruciski, John Wiley & Sons (1987), 7387.

....properties in G p ind for arbitrary edge probability functions p = p(n) one would need to know the ratio between H free subgraphs with a given number of edges and the number of all graphs with that number of edges. While this is in general not known for induced subgraphs the following result of [J LR90] provides such bounds for excluded weak subgraphs. Theorem 20. Let G n;p be a random graph with edge probability p = p(n) and let EH denote the event that G n;p contains no (weak) H subgraph. Then there exists a constant c H 0 such that Pr(EH ) exp( Gammac H min n n jAj p e(A) j A H; ....

Svante Janson, Thomasz / Luczak, and Andrezj Ruci'nski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, Random Graphs '87 (M. Karo'nski, J. Jaworski, and A. Ruci'nski, eds.), John Wiley & Sons, New York, 1990, pp. 73--87.

....of the presented approach for estimating the threshold for the existence of an H factor for other graphs H, in particular, for the case H = K r , r 3. 2 The Janson inequality In the course of the proof we will make a multiple use of the powerful inequality of Janson, first described in [4] (see also [1] Ch. 8) The following particular scheme of the inequality will suffice for our purposes. Let S be a family of labelled subgraphs of a complete graph on n labelled vertices. Each edge of this complete graph is chosen to be an edge of a random graph G 2 G(n; p) with probability p, ....

S. Janson, T. / Luczak and A. Ruci'nski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, in Random Graphs' 87 (M. Karo'nski et al., eds.), Wiley, Chichester, 1990, 73--87.

....desired bound. First, we compute as the number of all possible occurrences times the probability of one fixed occurrence. Remember that k is a constant. m k mk 2 k 2 k p 2(k 2 k) #(n k n k 2 k n 5 4 k 2 2(k 2 k) 2(k 2 k) #(n 5 4 ) Wecanbound#asin[18]by ## X ##=H#Gk 2 E(X(H,R p,m ) X(H,G k ) 2 , where the sum is over all graphs H with at least one edge which may occur as intersections G k (i) # G k (j)fori#=j. Since k is a constant, the number of such graphs is bounded and the values of X(H,G k ) are bounded as well, and therefore ....

S. Janson, T. Luczak, and A. Ruci nski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, in Random Graphs '87, M. Karonski et al., eds., John Wiley, New York, 1990, pp. 73--87.

.... occurrences times the probability of one fixed occurrence (remember that k is a constant) m k mk 2 k 2 Gamma k p 2(k 2 Gammak) Theta(n k n k 2 Gammak n 5=4 Gammak 2 2(k 2 Gammak) Delta2(k 2 Gammak) Theta(n 5=4 ) We can bound Delta as in [18] by Delta X ;6=HaeGk 2 E(X(H;R p;m ) X(H;G k ) 2 ; where the sum is over all graphs H with at least one edge which may occur as intersections G k (i) G k (j) for i 6= j. Since k is a constant, the number of such graphs is bounded and the values of X(H;G k ) are bounded as well, and ....

S. Janson, T. Luczak, and A. Ruci' nski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, in Random Graphs '87, M. K. et al., ed., Wilew, 1990, pp. 73--87.

....will be proven in the following manner: a) follows from Lemma 3.5 and Lemma 5.1, b) is Lemma 5.2, and (c) follows from Lemma 3.5, Lemma 5.3 and Property (P3) of G. A speci c value of , with which we apply Lemma 2. 4, is provided by a standard application of a correlation estimate from [13] (cf. 14] page 33) often called Janson s inequality. We say that a subgraph F G survives if F [ G(n; p) has a proper coloring in which F is monochromatic. Lemma 2.5 For every K 2 CORE, the probability that K survives is at most exp 2 0 n 3=2 0 = Proof: By Lemma ....

S.Janson, T. Luczak and A.Rucinski, An exponential bound for the probability of nonexistence of a speci ed subgraph of a random graph, in: Proceedings of Random Graphs'87, Wiley, Chichester (1990), 73-87.

....called strictly balanced graphs. The asymptotic normality of XG for a wide range of p (as long as pn 1 mG and n p) # #) was proven in [18] Next, it was shown that the lower tail of the distribution of XG decays exponentially in the expectation of the least expected subgraph of G, see [10] (P(XG = 0) and [9] the general case) Namely, let #H : n which is roughly the expected number of copies of H in G(n, p) Then, for all # (0, 1] with c # 0 depending on G and #, XG #)EXG # min . 1.1) Date: October 31, 2002. Research of the second author ....

....for as long as p n 1 mG . In another uninteresting case, when p is below the threshold n 1 mG , we have M # = #(1) our upper bound is #(1) and the lower bound is p #(1) On the other hand, at least for balanced G, the correct probability is just P(XG 0) which, by the result of [10], see (1.1) is #(EXG ) Indeed, when EXG 1 2, to have more than 2EXG copies of G is the same as to have at least one. The fact that the upper bound is not sharp here does not undermine our belief stated above. The important di#erence is that in this case, 1 is much more that 2EXG . Remark 8.2. ....

S. Janson, T. # Luczak & A. Rucinski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph. Random Graphs '87 (Proceedings, Poznan

....bounds are rather good. However, for p n 2 3 , they are pretty useless. Also for larger p, they are not sharp (unless p is bounded away from 0) Indeed, for this problem the following is known by other methods, assuming for simplicity that n # # and that p np 1. As is shown in [15], 16, Chapter 3] the sharp exponent in the lower tail estimate (4.10) is, for any fixed # 1, really of the order min(n , n p) For the upper tail (4.9) the sharp exponent is of the order n (possibly up to a factor log n) for any fixed # 0, see [17] See also partial results ....

....here for simplicity discussed the number of triangles. Similar results hold for the number of copies of any fixed graph in G(n, p) Again, Corollary 2. 6 gives estimates that are exponentially small for certain ranges of p, but we do not obtain optimal results (except in some extreme cases) cf. [15], 16] 22] 18] 17] Example 4.4 (Random hypergraphs) Another problem that was used in [18] to compare several di#erent methods was the following: Consider a fixed hypergraph H, for simplicity assumed to be uniform (all hyperedges have the same number of vertices) Delete vertices of H at ....

S. Janson, T. # Luczak & A. Rucinski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph. Random Graphs '87 (Proceedings, Poznan

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S.Janson, T./Luczak and A.Ruci'nski, An exponential bound for the probability of nonexistence of a specified subgraph of a random graph Proceedings of Random Graphs' 87 Wiley, Chichester 1990, 73-87.

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S. Janson, T. Luczak and A. Rucinski, An exponential bound for the probability of nonexistence of a specified subgraph in a random graph, in: M. Karonski et al. eds., Random graphs 87 (Wiley , New York, 1990) 73-87.

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