Title: Rainbow matchings in properly-coloured multigraphs,
Abstract: Aharoni and Berger conjectured that in any bipartite multigraph that is
properly edge-coloured by $n$ colours with at least $n + 1$ edges of each
colour there must be a matching that uses each colour exactly once. In this
paper we consider the same question without the bipartiteness assumption. We
show that in any multigraph with edge multiplicities $o(n)$ that is properly
edge-coloured by $n$ colours with at least $n + o(n)$ edges of each colour
there must be a matching of size $n-O(1)$ that uses each colour at most once.