A Vizing-type theorem for matching forests

A well-known Theorem of Vizing states that one can colour the edges of a graph by Δ+α colours, such that edges of the same colour form a matching. Here, Δ denotes the maximum degree of a vertex, and α the maximum multiplicity of an edge in the graph. An analogue of this Theorem for directed graphs was proved by Frank. It states that one can colour the arcs of a digraph by Δ+α colours, such that arcs of the same colour form a branching. For a digraph, Δ denotes the maximum indegree of a vertex, and α the maximum multiplicity of an arc. We prove a common generalization of the above two theorems concerning the colouring of mixed graphs (these are graphs having both directed and undirected edges) in such a way that edges of the same colour form a matching forest.