Threshold Functions for Asymmetric Ramsey Properties Involving Cliques

Marciniszyn, Martin; Skokan, Jozef; Spöhel, Reto; Steger, Angelika

doi:10.1007/11830924_42

Cited by 7 publications

(3 citation statements)

References 14 publications

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“…A considerable amount of technicalities in this section is specifically needed to deal with the case 2 = 3. For a simpler account of the case 2 ≥ 4, the reader is referred to [14]. Lastly, in Section 4, we explain how the 1-statement follows from the results in [9] and the K LR-Conjecture.…”

Section: Organization Of This Papermentioning

confidence: 98%

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: Organization Of This Papermentioning

confidence: 98%

Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn

Skokan

Spöhel

et al. 2008

Random Struct Algorithms

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“…In fact, this proof applies to all graphs F that have m 2 ( F ) ≥ 1. Our proof of the 0‐statement of Theorem 3 on the other hand follows a new approach by further developing algorithmic ideas from [10] to obtain a suitable coloring of G ( n , p ) (see section 3 in [10]) and combining them with a general theorem from [12] about the global density of graphs which are Ramsey with respect to a given graph, i.e., when \documentclass{article} \usepackage{amsmath,amsfonts,amssymb} \pagestyle{empty} \begin{document}$H \nrightarrow (F)^e_2$\end{document}, see Theorem 4.…”

Section: Introductionmentioning

confidence: 99%

Development of an industrial boiler virtual‐lab for control education using Modelica

Martin‐Villalba

Urquía

Dormido

2013

Comp Applic In Engineering

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ABSTRACT:The use of the VirtualLabBuilder Modelica library and the Dymola modeling environment facilitates the implementation of virtual-labs with elaborated user interfaces, and based on large and complex Modelica models. This implementation methodology is applied to develop an industrial boiler virtual-lab for process control education. VirtualLabBuilder is freely available at www.euclides.dia.uned.es.

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“…We shall consider the classical G n,p model of binomial random graphs, that is, G n,p consists of n labeled vertices, and the edges are independently present with probability p = p(n). We mention that several authors have investigated r g (G, P) and r l (G, P) in random or pseudorandom graphs for various properties P. For instance, the global resilience of G n,p with respect to the Turán property of containing a clique K t of a given order, or, more generally, the property of containing a given graph H of fixed order was studied in [18], [20], and [25]; for the case in which H is a cycle, see [10], [13], and [14] (for further related results, see [16], [17], and [21]). More recently, Sudakov and Vu [24] determined the local resilience of G n,p with respect to several properties, namely having a perfect matching, being Hamiltonian, being non-symmetric, and being k-colorable for a given function k = k(n).…”

Section: Introductionmentioning

confidence: 99%

On the Resilience of Long Cycles in Random Graphs

Dellamonica¹,

Kohayakawa²,

Marciniszyn³

et al. 2008

Electron. J. Combin.

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In this paper we determine the local and global resilience of random graphs $G_{n, p}$ ($p \gg n^{-1}$) with respect to the property of containing a cycle of length at least $(1-\alpha)n$. Roughly speaking, given $\alpha > 0$, we determine the smallest $r_g(G, \alpha)$ with the property that almost surely every subgraph of $G = G_{n, p}$ having more than $r_g(G, \alpha) |E(G)|$ edges contains a cycle of length at least $(1 - \alpha) n$ (global resilience). We also obtain, for $\alpha < 1/2$, the smallest $r_l(G, \alpha)$ such that any $H \subseteq G$ having $\deg_H(v)$ larger than $r_l(G, \alpha) \deg_G(v)$ for all $v \in V(G)$ contains a cycle of length at least $(1 - \alpha) n$ (local resilience). The results above are in fact proved in the more general setting of pseudorandom graphs.

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Threshold Functions for Asymmetric Ramsey Properties Involving Cliques

Abstract: Abstract. Consider the following problem: For given graphs G and F1, .

Cited by 7 publications

References 14 publications

Asymmetric Ramsey properties of random graphs involving cliques

Asymmetric Ramsey properties of random graphs involving cliques

Development of an industrial boiler virtual‐lab for control education using Modelica

On the Resilience of Long Cycles in Random Graphs

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