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    Linearly many rainbow trees in properly edge-coloured complete graphs

    Pokrovskiy, Alexey and Sudakov, B. (2019) Linearly many rainbow trees in properly edge-coloured complete graphs. Technical Report. Birkbeck College, University of London, London, UK.

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    A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjecture.


    Item Type: Monograph (Technical Report)
    Additional Information: Birkbeck Pure Mathematics Preprint Series #54
    School: Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School
    Depositing User: Administrator
    Date Deposited: 17 Jun 2020 09:48
    Last Modified: 02 Aug 2023 18:00


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