Clemens, D;
Ehrenmüller, J;
Pokrovskiy, A;
(2017)
On sets not belonging to algebras and rainbow matchings in graphs.
Journal of Combinatorial Theory, Series B
, 122
pp. 109-120.
10.1016/j.jctb.2016.05.006.
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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 Nivasch 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).
Type: | Article |
---|---|
Title: | On sets not belonging to algebras and rainbow matchings in graphs |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.1016/j.jctb.2016.05.006 |
Publisher version: | https://doi.org/10.1016/j.jctb.2016.05.006 |
Language: | English |
Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
Keywords: | Rainbow matchings, Edge colourings, Multigraphs, Equivalence classes |
UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
URI: | https://discovery.ucl.ac.uk/id/eprint/10105840 |
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