Full rainbow matchings in graphs and hypergraphs

TL;DR

This paper proves the existence of full rainbow matchings in certain edge-coloured graphs and hypergraphs, resolving an open problem and providing counterexamples to existing conjectures.

Contribution

It establishes conditions under which full rainbow matchings exist in graphs and hypergraphs, answering an open problem and extending previous conjectures.

Findings

Existence of full rainbow matchings under specified conditions

Counterexamples to several conjectures on rainbow matchings

Results applicable to multigraphs and hypergraphs

Abstract

Let $G$ be a simple graph that is properly edge coloured with $m$ colours and let $\M=\{M_1,\ldots, M_m\}$ be the set of $m$ matchings induced by the colours in $G$. Suppose that $m\le n-n^{c}$, where $c>9/10$, and every matching in $\M$ has size $n$. Then $G$ contains a full rainbow matching, i.e.\ a matching that contains exactly one edge from $M_i$ for each $1\le i\le m$. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalisation 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.