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Asymmetric Ramsey properties of random graphs involving cliques

Marciniszyn, Martin, Skokan, Jozef, Spöhel, Reto and Steger, Angelika (2009) Asymmetric Ramsey properties of random graphs involving cliques. Random Structures and Algorithms, 34 (4). pp. 419-453. ISSN 1042-9832

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Abstract

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 Ruciski 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- for some constants b = b(F,k) and = (F). This result is essentially best possible because for p Bn-, where B = B(F,k) is a large constant, such an edge-coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n-(F1,,Fk) 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- for some constant b = b(F1,,Fk), where = (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- the random graph Gn,p a.a.s. does not have a valid k-edge-coloring provided the so-called KR-conjecture holds. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Item Type: Article
Official URL: http://www3.interscience.wiley.com/journal/38107/h...
Additional Information: © 2009 Wiley Periodicals, Inc
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Departments > Mathematics
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Date Deposited: 24 Apr 2009 13:58
URL: http://eprints.lse.ac.uk/23749/

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