Results 1  10
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34
An approximate version of Sidorenko’s conjecture
 Geom. Funct. Anal
"... A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent ana ..."
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Cited by 11 (3 self)
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A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs. 1
Nonbacktracking random walks mix faster
, 2006
"... We compute the mixing rate of a nonbacktracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the ..."
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Cited by 9 (3 self)
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We compute the mixing rate of a nonbacktracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a highgirth regular expander on n vertices, then a typical nonbacktracking random walk of length n on G does not visit a vertex more than log n (1 + o(1)) log log n times, and this result is tight. In this sense, the multiset of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times. 1
Mixing time of exponential random graphs
, 2008
"... A plethora of random graph models have been developed in recent years to study a range of problems on networks, driven by the wide availability of data from many social, telecommunication, biochemical and other networks. A key model, extensively used in the sociology literature, is the exponential r ..."
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Cited by 9 (0 self)
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A plethora of random graph models have been developed in recent years to study a range of problems on networks, driven by the wide availability of data from many social, telecommunication, biochemical and other networks. A key model, extensively used in the sociology literature, is the exponential random graph model. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for answering almost any problem of parameter estimation hypothesis testing or to understand the inherent network model itself. In practice this sampling is typically carried out using either the Glauber dynamics or the MetropolisHasting Markov chain Monte Carlo procedure. In this paper we characterize the high and low temperature regimes. We establish that in the high temperature regime the mixing time of the Glauber dynamics is Θ(n 2 log n), where n is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the ErdősRényi random graph. 1
On packing Hamilton Cycles in ɛRegular Graphs
, 2003
"... A graph G = (V; E) on n vertices is (; )regular if its minimal degree is at least n, and for every pair of disjoint subsets S; T V of cardinalities at least n, the number of edges e(S; T ) between S and T satis es: e(S;T ) . We prove that if > 0 are constants, then every (; )regular ..."
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Cited by 7 (6 self)
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A graph G = (V; E) on n vertices is (; )regular if its minimal degree is at least n, and for every pair of disjoint subsets S; T V of cardinalities at least n, the number of edges e(S; T ) between S and T satis es: e(S;T ) . We prove that if > 0 are constants, then every (; )regular graph on n vertices contains a family of (=2 O())n edgedisjoint Hamilton cycles. As a consequence we derive that for every constant 0 < p < 1, with high probability in the random graph G(n; p), almost all edges can be packed into edgedisjoint Hamilton cycles. A similar result is proven for the directed case.
Local resilience and Hamiltonicity MakerBreaker games in random regular graphs
 Combinatorics, Probability, and Computing
"... For an increasing monotone graph property P the local resilience of a graph G with respect to P is the minimal r for which there exists of a subgraph H ⊆ G with all degrees at most r such that the removal of the edges of H from G creates a graph that does not possesses P. This notion, which was impl ..."
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Cited by 6 (2 self)
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For an increasing monotone graph property P the local resilience of a graph G with respect to P is the minimal r for which there exists of a subgraph H ⊆ G with all degrees at most r such that the removal of the edges of H from G creates a graph that does not possesses P. This notion, which was implicitly studied for some adhoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model G(n, p) and some families of pseudorandom graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudorandom regular graphs of constant degree. We investigate the local resilience of the typical random dregular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive ε and large enough values of d with high probability the local resilience of the random dregular graph, Gn,d, with respect to being Hamiltonian is at least (1−ε)d/6. We also prove that for the Binomial random graph model G(n, p), for every positive ε> 0 and large enough values of K, if p> K ln n n then with high probability the local resilience of G(n, p) with respect to being Hamiltonian is at least (1 − ε)np/6. Finally, we apply similar techniques to Positional Games and prove that if d is large enough then with high probability a typical random dregular graph G is such that in the unbiased MakerBreaker game played on the edges of G, Maker has a winning strategy to create a Hamilton cycle. 1
Random regular graphs of nonconstant degree: edge distribution and applications
, 2006
"... In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model Gn,d for d = o(n 1/5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that ..."
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Cited by 5 (4 self)
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In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model Gn,d for d = o(n 1/5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G(n, p) with p = d. Our proof is largely based on ideas of Alon n and Krivelevich who proved this twopoint concentration result for G(n, p) for p = n −δ where δ> 1/2. The main tool used to derive such a result is a careful analysis of the distribution of edges in Gn,d, relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model. 1
Induced Ramseytype theorems
, 2008
"... We present a unified approach to proving Ramseytype theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rödl, Erdős–Hajnal, Prömel–Rödl, Nikiforov, Chung–Graham, and Łuczak–Rödl. The proofs are based on a simple lemma (generalizing ..."
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Cited by 4 (4 self)
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We present a unified approach to proving Ramseytype theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rödl, Erdős–Hajnal, Prömel–Rödl, Nikiforov, Chung–Graham, and Łuczak–Rödl. The proofs are based on a simple lemma (generalizing one by Graham, Rödl, and Ruciński) that can be used as a replacement for Szemerédi’s regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudorandom graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.
kwise independent random graphs
 2008 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS
"... We study the kwise independent relaxation of the usual model G(N, p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be r ..."
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Cited by 3 (1 self)
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We study the kwise independent relaxation of the usual model G(N, p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be relevant in modeling phenomena where only kwise independence is assumed to hold, and is also useful when the relevant graphs are so huge that handling G(N, p) graphs becomes infeasible, and cheaper randomlooking distributions (such as kwise independent ones) must be used instead. Unfortunately, many wellknown properties of random graphs in G(N, p) are global, and it is thus not clear if they are guaranteed to hold in the kwise independent case. We explore the properties of kwise independent graphs by providing upperbounds and lowerbounds on the amount of independence, k, required for maintaining the main properties of G(N, p) graphs: connectivity, Hamiltonicity, the connectivitynumber, cliquenumber and chromaticnumber and the appearance of fixed subgraphs. Most of these properties are shown to be captured by either constant k or by some k = poly(log(N)) for a wide range of values of p, implying that random looking graphs on N vertices can be generated by a seed of size poly(log(N)). The proofs combine combinatorial, probabilistic and spectral techniques. 1.