1-factor covers of regular graphs

We consider minimal 1-factor covers of regular multigraphs, focusing on those that are 1-factorizations. In particular, we classify cubic graphs such that every minimal 1-factor cover is also a 1-factorization, and also classify simple regular bipartite graphs with this property. For r>3r>3, we show that there are finitely many simple rr-regular graphs such that every minimal 1-factor cover is also a 1-factorization.