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    Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

    Alon, N. and Pokrovskiy, Alexey and Sudakov, B. (2017) Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles. Israel Journal of Mathematics 222 (1), pp. 317-331. ISSN 0021-2172.

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    Abstract

    A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured Kn has a rainbow cycle of length n−O(n^3/4). One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured Kn formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties.

    Metadata

    Item Type: Article
    Additional Information: The final publication is available at Springer via the link above.
    School: School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Alexey Pokrovskiy
    Date Deposited: 12 Dec 2018 16:53
    Last Modified: 24 Jun 2020 17:07
    URI: https://eprints.bbk.ac.uk/id/eprint/25438

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