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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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 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 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/25896 |
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