  In the Properly Colored Spanning Tree problem, we are given an edge-colored
undirected graph and the goal is to find a properly colored spanning tree,
i.e., a spanning tree in which any two adjacent edges have distinct colors. The
problem is interesting not only from a graph coloring point of view, but is
also closely related to the Degree Bounded Spanning Tree and (1,2)-Traveling
Salesman problems, two classical questions that have attracted considerable
interest in combinatorial optimization and approximation theory. Previous work
on properly colored spanning trees has mainly focused on determining the
existence of such a tree and hence has not considered the question from an
algorithmic perspective. We propose an optimization version called Maximum-size
Properly Colored Forest problem, which aims to find a properly colored forest
with as many edges as possible. We consider the problem in different graph
classes and for different numbers of colors, and present polynomial-time
approximation algorithms as well as inapproximability results for these
settings. Our proof technique relies on the sum of matching matroids defined by
the color classes, a connection that might be of independent combinatorial
interest.
  We also consider the Maximum-size Properly Colored Tree problem, which asks
for the maximum size of a properly colored tree not necessarily spanning all
the vertices. We show that the optimum is significantly more difficult to
approximate than in the forest case, and provide an approximation algorithm for
complete multigraphs.
