Szemerédi’s Regularity Lemma for Sparse Graphs

Kohayakawa, Yoshiharu

doi:10.1007/978-3-642-60539-0_16

Cited by 133 publications

(153 citation statements)

References 28 publications

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“…A standard way of attacking the 1-statement, which was also pursued in [9], is via the sparse version of Szemerédi's regularity lemma, which was independently developed by Kohayakawa [8] and Rödl (unpublished, see [11]). Using properties of regularity, one can find a monochromatic copy of a forbidden subgraph in the colored graph G n,p .…”

Section: Theorem 4 (Main Resultsmentioning

confidence: 99%

Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn

Skokan

Spöhel

et al. 2008

Random Struct Algorithms

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Consider the following problem: For given graphs G and F1,…,Fk, find a coloring of the edges of G with k colors such that G does not contain Fi in color i. Rödl and Ruciński studied this problem for the random graph Gn,p in the symmetric case when k is fixed and F1 = ··· = Fk = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p ≤ bn−β for some constants b = b(F,k) and β = β(F). This result is essentially best possible because for p ≥ Bn−β, where B = B(F,k) is a large constant, such an edge‐coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n for arbitrary F1,…,Fk.In this article we address the case when F1,…,Fk are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k‐edge‐coloring of Gn,p with p ≤ bn−β for some constant b = b(F1,…,Fk), where β = β(F1,…,Fk) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F1,…,Fk) such that for p ≥ Bn−β the random graph Gn,p a.a.s. does not have a valid k‐edge‐coloring provided the so‐called KŁR‐conjecture holds. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

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Section: Theorem 4 (Main Resultsmentioning

confidence: 99%

Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn

Skokan

Spöhel

et al. 2008

Random Struct Algorithms

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“…More importantly, such results turned out to be quite important in dealing with certain extremal and Ramsey type problems involving subgraphs of random graphs. The interested reader is referred to [36].…”

Section: Introductionmentioning

confidence: 99%

Szemerédi’s Regularity Lemma and Quasi-randomness

Kohayakawa¹,

Rödl²

2003

Recent Advances in Algorithms and Combinatorics

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Abstract. The first half of this paper is mainly expository, and aims at introducing the regularity lemma of Szemerédi. Among others, we discuss an early application of the regularity lemma that relates the notions of universality and uniform distribution of edges, a form of 'pseudorandomness' or 'quasi-randomness'. We then state two closely related variants of the regularity lemma for sparse graphs and present a proof for one of them.In the second half of the paper, we discuss a basic idea underlying the algorithmic version of the original regularity lemma: we discuss a 'local' condition on graphs that turns out to be, roughly speaking, equivalent to the regularity condition of Szemerédi. Finally, we show how the sparse version of the regularity lemma may be used to prove the equivalence of a related, local condition for regularity. This new condition turns out to give a O(n 2 ) time algorithm for testing the quasi-randomness of an n-vertex graph.

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“…Suppose that D(v) =d for some numberd =d(n) = o(n). Then the above notions of regularity and boundedness coincide with those of the "sparse regularity lemma" of Kohayakawa [21] and Rödl (unpublished). Hence, Theorem 1.2 provides an algorithmic version of this regularity concept.…”

Section: If We Let D(v)mentioning

confidence: 72%

“…This result has not been published previously (although it may have been known to experts in the area that this can be derived from Alon and Naor [4]-see, e.g., [16]). Actually devising an algorithm for computing a sparse regular partition is mentioned as an open problem in [21]. 3.…”

Section: If We Let D(v)mentioning

confidence: 99%

Quasi-randomness and Algorithmic Regularity for Graphs with General Degree Distributions

Alon

Coja-Oghlan

Hàn

et al.

Automata, Languages and Programming

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Abstract. We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph "resembles" a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if G = (V, E) satisfies a certain boundedness condition, then G admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72-80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph G in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without "dense spots."

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Szemerédi’s Regularity Lemma for Sparse Graphs

Cited by 133 publications

References 28 publications

Asymmetric Ramsey properties of random graphs involving cliques

Asymmetric Ramsey properties of random graphs involving cliques

Szemerédi’s Regularity Lemma and Quasi-randomness

Quasi-randomness and Algorithmic Regularity for Graphs with General Degree Distributions

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