Balanced list edge-colourings of bipartite graphs

Galvin solved the Dinitz conjecture by proving that bipartite graphs are ∆edge-choosable. We improve Galvin’s method and deduce from any colouring of the edges of bipartite graph G some further list edge-colouring properties of G. In particular, for bipartite graphs, it follows from the existence of balanced bipartite edge-colourings that balanced list edge-colourings exist as well. While the key to Galvin’s proof is the stable marriage theorem of Gale and Shapley, our result is based on the well-known “many-to-many” version of the stable matching theorem.