Quasi-random graphs

Chung, Fan; Graham, Ronald L.; Wilson, R.

doi:10.1007/bf02125347

Cited by 357 publications

(193 citation statements)

References 9 publications

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“…This is very surprising, since the properties may appear completely unrelated to one another. Quasirandom graphs, hypergraphs, set systems, subsets of Z N , and tournaments have been examined (see [5], [6] and [7] for details).…”

Section: K)mentioning

confidence: 99%

A note on the pseudorandomness of the Liouville function

Liu

Zhai

2009

Acta Arith.

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Section: K)mentioning

confidence: 99%

A note on the pseudorandomness of the Liouville function

Liu

Zhai

2009

Acta Arith.

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“…The second subsection discusses quasirandomness of graphs, a concept introduced by Chung, Graham and Wilson [3] (see also Thomason [31] for a similar notion). There are many ways of describing this concept, all of which are broadly equivalent (up to renaming constants); in fact, it is also equivalent to regularity (as described in the first subsection).…”

Section: Graphs: Regularity and Counting Quasirandomness And Quasirmentioning

confidence: 99%

“…Suppose G is a tripartite graph with parts V 1 , V 2 , V 3 Four-cycles. Suppose G = (X, Y ) is an -regular bipartite graph with density d. Let C 4 (G) be the number of labelled 4-cycles in G, i.e.…”

Section: Regularity and Countingmentioning

confidence: 99%

A hypergraph regularity method for generalized Turán problems

Keevash

2008

Random Struct Algorithms

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ABSTRACT:We describe a method that we believe may be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that is, any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2 − q + 1 and n/2, but for PG 2 (4) we find the somewhat surprising phenomenon that the threshold is less than (1/2 − )n for some small > 0. We conclude by setting out a program for future developments of this method to tackle other problems.

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“…A weaker (yet very useful) definition is quasi-random graphs, which requires only that ∀U |E(U )| − p |U | 2 ≤ o(N 2 ). Quasi-random graphs were shown by Chung, Graham and Wilson ( [6]) to be equivalent to the surprisingly innocent condition that the number of labeled cycles of length 4 is (pN ) 4 (1±o(1)) when E(g n ) = (p±o(1)) N 2 . Several deterministic constructions are known for such quasi-random and jumbled graphs (see a recent survey by Krivelevich and Sudakov [16]).…”

mentioning

confidence: 99%

“…By Theorem 9(a), we can deterministically compute an irreducible polynomial of degree n in GF (3)[X] in time poly(n ) = poly(n), and can thus efficiently calculate in the field F = GF(3 n ). 6 To deterministically find an element of order M in time poly(n), run the algorithm of Theorem 9(b) and, for each output element β, directly test whether γ = β (N −1)/S has order M by computing the first M powers of γ. Note that when β generates F * , γ indeed has order M .…”

mentioning

confidence: 99%

Efficiently Constructible Huge Graphs That Preserve First Order Properties of Random Graphs

Naor

Nussboim

Tromer

2005

Theory of Cryptography

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Abstract. We construct efficiently computable sequences of randomlooking graphs that preserve properties of the canonical random graphs G(2 n , p(n)). We focus on first-order graph properties, namely properties that can be expressed by a formula φ in the language where variables stand for vertices and the only relations are equality and adjacency (e.g. having an isolated vertex is a first-order property ∃x∀y(¬edge(x, y))). Random graphs are known to have remarkable structure w.r.t. first order properties, as indicated by the following 0/1 law: for a variety of choices of p(n), any fixed first-order property φ holds for G(2 n , p(n)) with probability tending either to 0 or to 1 as n grows to infinity.We first observe that similar 0/1 laws are satisfied by G(2 n , p(n)) even w.r.t. sequences of formulas {φn} n∈N with bounded quantifier depth, depth(φn) ≤ n lg(1/p(n)) . We also demonstrate that 0/1 laws do not hold for random graphs w.r.t. properties of significantly larger quantifier depth. For most choices of p(n), we present efficient constructions of huge graphs with edge density nearly p(n) that emulate G(2 n , p(n)) by satisfying Θ( n lg(1/p(n)) )-0/1 laws. We show both probabilistic constructions (which also have other properties such as K-wise independence and being computationally indistinguishable from G(N, p(n)) ), and deterministic constructions where for each graph size we provide a specific graph that captures the properties of G(2 n , p(n)) for slightly smaller quantifier depths.

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Quasi-random graphs

Cited by 357 publications

References 9 publications

A note on the pseudorandomness of the Liouville function

A note on the pseudorandomness of the Liouville function

A hypergraph regularity method for generalized Turán problems

Efficiently Constructible Huge Graphs That Preserve First Order Properties of Random Graphs

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