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Full rainbow matchings in graphs and hypergraphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  20 January 2021

Pu Gao ,
Reshma Ramadurai ,
Ian M. Wanless  and
Nick Wormald
Pu Gao
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Ontario, N2L 3G1, Canada
Reshma Ramadurai
Affiliation:
School of Mathematics & Statistics, Victoria University of Wellington, Wellington6140, New Zealand
Ian M. Wanless
Affiliation:
School of Mathematics, Monash University, Victoria3800, Australia
Nick Wormald*
Affiliation:
School of Mathematics, Monash University, Victoria3800, Australia
*
*Corresponding author. Email: nick.wormald@monash.edu
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Abstract

Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\], where \[c > 9/10\], and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from Mi for each \[1 \leqslant i \leqslant m\]. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs.

Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

MSC classification

Primary: 05D15: Transversal (matching) theory
Secondary: 05C70: Factorization, matching, partitioning, covering and packing
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 762 - 780
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

Research supported by the ARC grant DE170100716. The research was conducted when the author was affiliated with Monash University.

‡

Research supported by the ARC grant DP150100506.

§

Research supported by the Australian Laureate Fellowships grant FL120100125.

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