BIROn - Birkbeck Institutional Research Online

    On sets not belonging to algebras and rainbow matchings in graphs

    Clemens, D. and Ehrenmüller, J. and Pokrovskiy, Alexey (2016) On sets not belonging to algebras and rainbow matchings in graphs. Journal of Combinatorial Theory, Series B 122 , pp. 109-120. ISSN 0095-8956.

    [img]
    Preview
    Text
    1508.06437.pdf - Author's Accepted Manuscript
    Available under License Creative Commons Attribution Non-commercial No Derivatives.

    Download (377kB) | Preview

    Abstract

    Motivated by a question of Grinblat, we study the minimal number v(n) that satisfies the following. If A1,…,An are equivalence relations on a set X such that for every i∈[n] there are at least v(n) elements whose equivalence classes with respect to Ai are nontrivial, then A1,…,An contain a rainbow matching, i.e. there exist 2n distinct elements x1,y1,…,xn,yn∈X with xi∼Aiyi for each i∈[n]. Grinblat asked whether v(n)=3n−2 for every n≥4. The best-known upper bound was v(n)≤16n/5+O(1) due to Nivash and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that v(n)=3n+o(n).

    Metadata

    Item Type: Article
    Keyword(s) / Subject(s): Rainbow matchings, Edge colourings, Multigraphs, Equivalence classes
    School: Birkbeck Schools and Departments > School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Alexey Pokrovskiy
    Date Deposited: 21 Jan 2019 11:17
    Last Modified: 28 Jul 2019 18:53
    URI: http://eprints.bbk.ac.uk/id/eprint/25896

    Statistics

    Downloads
    Activity Overview
    40Downloads
    23Hits

    Additional statistics are available via IRStats2.

    Archive Staff Only (login required)

    Edit/View Item Edit/View Item