  One of the most important questions in matroid optimization is to find
disjoint common bases of two matroids. The significance of the problem is
well-illustrated by the long list of conjectures that can be formulated as
special cases. B\'erczi and Schwarcz showed that the problem is hard in
general, therefore identifying the borderline between tractable and intractable
instances is of interest.
  In the present paper, we study the special case when one of the matroids is a
partition matroid while the other one is a graphic matroid. This setting is
equivalent to the problem of packing rainbow spanning trees, an extension of
the problem of packing arborescences in directed graphs which was answered by
Edmonds' seminal result on disjoint arborescences. We complement his result by
showing that it is NP-complete to decide whether an edge-colored graph contains
two disjoint rainbow spanning trees. Our complexity result holds even for the
very special case when the graph is the union of two spanning trees and each
color class contains exactly two edges. As a corollary, we give a negative
answer to a question on the decomposition of oriented $k$-partition-connected
digraphs.
