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Properly coloured copies and rainbow copies of large graphs with small maximum degree

Böttcher, Julia, Kohayakawa, Yoshiharu and Procacci, Aldo (2011) Properly coloured copies and rainbow copies of large graphs with small maximum degree. Random Structures and Algorithms, 40 (4). pp. 425-436. ISSN 1042-9832

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Abstract

Let G be a graph on n vertices with maximum degree Δ. We use the Lovász local lemma to show the following two results about colourings χ of the edges of the complete graph Kn. If for each vertex v of Kn the colouring χ assigns each colour to at most (n - 2)/(22.4Δ2) edges emanating from v, then there is a copy of G in Kn which is properly edge-coloured by χ. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409–433, 2003]. On the other hand, if χ assigns each colour to at most n/(51Δ2) edges of Kn, then there is a copy of G in Kn such that each edge of G receives a different colour from χ. This proves a conjecture of Frieze and Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a framework developed by Lu and Székely [Electron. J. Comb. 14(1), R63, 2007] for applying the local lemma to random injections. In order to improve the constants in our results we use a version of the local lemma due to Bissacot, Fernández, Procacci, and Scoppola [preprint, arXiv:0910.1824]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 425–436, 2012

Item Type: Article
Official URL: http://onlinelibrary.wiley.com/doi/10.1002/rsa.203...
Additional Information: © 2011 Wiley Periodicals
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: 08 Aug 2012 12:47
URL: http://eprints.lse.ac.uk/45256/

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