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    Rainbow matchings and rainbow connectedness

    Pokrovskiy, Alexey (2017) Rainbow matchings and rainbow connectedness. Electronic Journal of Combinatorics 24 (1), pp. 1-13. ISSN 1077-8926.

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    Abstract

    Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. In the case when the matchings are much larger than n + 1, the best bound is currently due to Clemens and Ehrenmüller who proved the conjecture when the matchings are of size at least 3n/2 + o(n). When the matchings are all edge-disjoint and perfect, then the best result follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least n + o(n). In this paper we show that the conjecture is true when the matchings have size n + o(n) and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least φn+o(n) where φ≈1.618 is the Golden Ratio. Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

    Metadata

    Item Type: Article
    School: School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Alexey Pokrovskiy
    Date Deposited: 21 Jan 2019 10:44
    Last Modified: 11 Sep 2020 05:56
    URI: http://eprints.bbk.ac.uk/id/eprint/25891

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