Rainbow matchings in bipartite multigraphs

Abstract

Suppose that k is a non-negative integer and a bipartite multigraph G is the union of $$\begin{aligned} N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1) \end{aligned}$$ matchings \(M_1,\dots ,M_N\), each of size n. We show that G has a rainbow matching of size \(n-k\), i.e. a matching of size \(n-k\) with all edges coming from different \(M_i\)’s. Several choices of the parameter k relate to known results and conjectures.