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    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.

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    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).


    Item Type: Article
    Keyword(s) / Subject(s): Rainbow matchings, Edge colourings, Multigraphs, Equivalence classes
    School: Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School
    Depositing User: Alexey Pokrovskiy
    Date Deposited: 21 Jan 2019 11:17
    Last Modified: 02 Aug 2023 17:47


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