Partitioning to three matchings of given size is NP-complete for bipartite graphs

We show that the problem of deciding whether the edge set of a bipartite graph can be partitioned into three matchings, of size k₁, k₂ and k₃ is NP-complete, even if one of the matchings is required to be perfect. We also show that the problem of deciding whether the edge set of a simple graph contains a perfect matching and a disjoint matching of size k or not is NP-complete, already for bipartite graphs with maximum degree 3. It also follows from our construction that it is NP-complete to decide whether in a bipartite graph there is a perfect matching and a disjoint matching that covers all vertices whose degree is at least 2.