A vizing-type theorem for matching forests

A well known Theorem of Vizing states that one can colour the edges of a graph by $\Delta +\alpha$ colours, such that edges of the same colour form a matching. Here, $\Delta$ denotes the maximum degree of a vertex, and $\alpha$ 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 $\Delta +\alpha$ colours, such that arcs of the same colour form a branching. For a digraph, $\Delta$ denotes the maximum indegree of a vertex, and $\alpha$ the maximum multiplicity of an arc.