OnK 4-free subgraphs of random graphs

Kohayakawa, Yoshiharu; Łuczak, Tomasz; Rödl, Vojtěch

doi:10.1007/bf01200906

Cited by 117 publications

(153 citation statements)

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“…Using properties of regularity, one can find a monochromatic copy of a forbidden subgraph in the colored graph G n,p . Unfortunately, generalizing this argument from cycles to cliques requires a proof of Conjecture 23 in [10] (cf. Conjecture 30 below) of Kohayakawa, Luczak, and Rödl.…”

Section: Theorem 4 (Main Resultsmentioning

confidence: 99%

“…Suppose 0 <p ≤ 1 and 0 < ε ≤ 1 are real numbers, and U and W are disjoint nonempty subsets of V . Define thep-density of F between U and W by This lemma is a special case of the following conjecture that appeared in [10]. Note that the factors ( + 2) and 2 respectively in the binomial coefficients are negligible since they contribute only a factor of O(1) T to the total expression, which may be suppressed by choosing α sufficiently small.…”

Section: The 1-statementmentioning

confidence: 99%

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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%

Section: The 1-statementmentioning

confidence: 99%

Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn

Skokan

Spöhel

et al. 2008

Random Struct Algorithms

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“…The systematic study of extensions of (1) arising from replacing K n with a sparse random or a pseudorandom graph was initiated by Kohayakawa and collaborators (see, e.g., [12,13,15,16], see also [6] for some earlier work for Gpn, 1{2q). For random graphs such extensions were obtained recently in [9,19].…”

Section: §1 Introduction and Main Resultsmentioning

confidence: 99%

Extremal results for odd cycles in sparse pseudorandom graphs

Aigner-Horev

Hàn²,

Schacht³

2014

Combinatorica

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Abstract. We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Γ the generalized Turán density π F pΓq denotes the density of a maximum subgraph of Γ, which contains no copy of F . Extending classical Turán type results for odd cycles, we show that π F pΓq " 1{2 provided F is an odd cycle and Γ is a sufficiently pseudorandom graph.In particular, for pn, d, λq-graphs Γ, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval r´λ, λs, our result holds for odd cycles of length , providedUp to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabó, and where χpHq denotes the chromatic number of H.The systematic study of extensions of (1) arising from replacing K n with a sparse random or a pseudorandom graph was initiated by Kohayakawa and collaborators (see, E. Aigner-Horev

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“…Using properties of regularity, one can find a monochromatic copy of a forbidden subgraph in the colored graph G n,p . Unfortunately, generalizing this argument from cycles to cliques requires a proof of Conjecture 23 in [11] of Kohayakawa, Luczak, and Rödl. This so-called K LR-Conjecture formulates a probabilistic version of the classical embedding lemma for dense graphs.…”

Section: Conjecture 3 ([9]mentioning

confidence: 99%