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23
An optimal algorithm for checking regularity
 SIAM J. ON COMPUTING
"... We present a deterministic algorithm A that, in O(m 2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemerédi [E. Szemerédi, Regular partitions of graphs, Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Univ. Orsay, ..."
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Cited by 23 (6 self)
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We present a deterministic algorithm A that, in O(m 2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemerédi [E. Szemerédi, Regular partitions of graphs, Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Univ. Orsay,
Weak hypergraph regularity and linear hypergraphs
, 2009
"... We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any kuniform hypergraph H of positive uniform density contains all linear kuniform hypergraphs of a given size. More precisely, we show that for all integers ℓ ≥ k ≥ 2 and every d&g ..."
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Cited by 20 (6 self)
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We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any kuniform hypergraph H of positive uniform density contains all linear kuniform hypergraphs of a given size. More precisely, we show that for all integers ℓ ≥ k ≥ 2 and every d> 0 there exists ϱ> 0 for which the following holds: if H is a sufficiently large kuniform hypergraph with the property that the density of H induced on every vertex subset of size ϱn is at least d, then H contains every linear kuniform hypergraph F with ℓ vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph εregularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems.
A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring
, 2003
"... Let R be the set of all finite graphs G with the Ramsey property that every coloring of the edges of G by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let G(n, p) be the random graph on n vertices with edge probability ..."
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Cited by 19 (4 self)
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Let R be the set of all finite graphs G with the Ramsey property that every coloring of the edges of G by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let G(n, p) be the random graph on n vertices with edge probability p. We prove that there exists a function �c = �c(n) with 0 <c<�c<Csuch that for any ε>0, as n tends to infinity and Pr � G(n, (1 − ε)�c / √ n) ∈R � →0 Pr � G(n, (1 + ε)�c / √ n) ∈R � →1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemerédi’s Regularity Lemma to a certain hypergraph setting.
Extremal results in sparse pseudorandom graphs
 Adv. Math. 256 (2014), 206–290. arXiv:1204.6645 doi:10.1016/j.aim.2013.12.004 MR3177293
"... Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extendin ..."
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Cited by 13 (9 self)
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Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi’s regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a wellknown open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several wellknown combinatorial theorems, including the removal lemmas for graphs and groups, the ErdősStoneSimonovits theorem and Ramsey’s
Sparse graphs: metrics and random models
"... Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the d ..."
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Cited by 12 (1 self)
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Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for largescale realworld networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasirandom graphs are in a sense completely
Grothendiecktype inequalities in combinatorial optimization
 Comm. Pure Appl. Math
, 2011
"... Abstract. We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. Contents ..."
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Cited by 11 (4 self)
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Abstract. We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. Contents
SZEMERÉDI’S REGULARITY LEMMA FOR MATRICES AND SPARSE GRAPHS
"... Abstract. Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that ho ..."
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Cited by 9 (0 self)
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Abstract. Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices. 1.
AN Lp THEORY OF SPARSE GRAPH CONVERGENCE I: LIMITS, SPARSE RANDOM GRAPH MODELS, AND POWER LAW
"... Abstract. We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L ∞ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots ..."
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Cited by 8 (1 self)
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Abstract. We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L ∞ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the Lp theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.
REGULARITY LEMMAS FOR GRAPHS
"... Szemerédi’s regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some applications variants of the regularity lemma were considered. Here we discuss several of those variants an ..."
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Cited by 7 (1 self)
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Szemerédi’s regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some applications variants of the regularity lemma were considered. Here we discuss several of those variants and their relation to each other.