Graphs with Monochromatic Complete Subgraphs in Every Edge Coloring

Folkman, Jon

doi:10.1137/0118004

Cited by 154 publications

(100 citation statements)

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“…As in (2), it is useful to denote the maximization of this generalized 2-density over all subgraphs J ⊆ F with respect to some graph H by m 2 (H, F ). The graph F is balanced w.r.t.…”

Section: Preliminaries and Notationmentioning

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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“…As in (2), it is useful to denote the maximization of this generalized 2-density over all subgraphs J ⊆ F with respect to some graph H by m 2 (H, F ). The graph F is balanced w.r.t.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn

Skokan

Spöhel

et al. 2008

Random Struct Algorithms

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show abstract

“…The existence of a K 4 -free graph G which is t-Ramsey for K 3 was settled by Folkman [53] (t = 2) and Nešetřil and Rödl [94]. The proofs are complicated and the graphs constructed are very large.…”

Section: Theorem 6 ([31]mentioning

confidence: 99%

Ramsey Theory in the Work of Paul Erdős

Graham¹,

Nešetřil²

1997

Algorithms and Combinatorics

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“…Besides the number of vertices, the notion has been studied with ρ being the clique number [4,10], chromatic number [2,14], number of edges (yielding the "size Ramsey number") [3,11], and maximum degree [8,9]. In this paper we let ρ be the "circular chromatic number".…”

Section: Introductionmentioning

confidence: 99%