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13
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 17 (4 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> ..."
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Cited by 12 (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 11 (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.
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 8 (0 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
REGULARITY LEMMAS FOR GRAPHS
"... Abstract. 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 var ..."
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Cited by 3 (1 self)
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Abstract. 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. 1.
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 3 (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.
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 3 (3 self)
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Abstract. We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. Contents
Ramsey games against a onearmed bandit
 COMBIN. PROBAB. COMPUT
, 2003
"... We study the following oneperson game against a random graph: the Player's goal is to 2colour a random sequence of edges e1, e2,... of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online v ..."
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Cited by 2 (0 self)
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We study the following oneperson game against a random graph: the Player's goal is to 2colour a random sequence of edges e1, e2,... of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge when it comes, before looking at the next edge. While it is not hard to prove that the expected length of this game is about n