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

    Noga, A. and Pokrovskiy, Alexey and Sudakov, B. (2018) Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles. Technical Report. Birkbeck College, University of London, London, UK.

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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 the subgraph of a properly edge-coloured Kn formed by the edges 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: Monograph (Technical Report)
    Additional Information: Birkbeck Pure Mathematics Preprint Series #55
    School: School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Administrator
    Date Deposited: 17 Jun 2020 09:51
    Last Modified: 16 Feb 2021 07:43
    URI: https://eprints.bbk.ac.uk/id/eprint/32283

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