Abstract. Recently, it was proved by Bérczi and Schwarcz that the problem of factorizing a matroid into rainbow bases with respect to a given partition of its ground set is algorithmically intractable. On the other hand, many special cases were left open. We first show that the problem remains hard if the matroid is graphic, answering a question of Bérczi and Schwarcz. As another special case, we consider the problem of deciding whether a given digraph can be factorized into subgraphs which are spanning trees in the underlying sense and respect upper bounds on the indegree of every vertex. We prove that this problem is also hard. This answers a question of Frank. In the second part of the article, we deal with the relaxed problem of covering the ground set of a matroid by rainbow bases. Among other results, we show that there is a linear function [Formula: see text] such that every matroid that can be factorized into [Formula: see text] bases for some [Formula: see text] can be covered by [Formula: see text] rainbow bases if every partition class contains at most 2 elements.

It is proved that if a finite graph G has a factorization into two spanning trees then it admits one such that all leaves in either factor have degree at most 8 in G. © 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 57(4), 351-353 2011