  In this paper we study the oriented vertex and arc coloring problem on edge
series-parallel digraphs (esp-digraphs) which are related to the well known
series-parallel graphs. Series-parallel graphs are graphs with two
distinguished vertices called terminals, formed recursively by parallel and
series composition. These graphs have applications in modeling series and
parallel electric circuits and also play an important role in theoretical
computer science. The oriented class of series-parallel digraphs is recursively
defined from pairs of vertices connected by a single arc and applying the
parallel and series composition, which leads to specific orientations of
undirected series-parallel graphs. Further we consider the line digraphs of
edge series-parallel digraphs, which are known as minimal series-parallel
digraphs (msp-digraphs).
  We show tight upper bounds for the oriented chromatic number and the oriented
chromatic index of edge series-parallel digraphs and minimal series-parallel
digraphs. Furthermore, we introduce first linear time solutions for computing
the oriented chromatic number of edge series-parallel digraphs and the oriented
chromatic index of minimal series-parallel digraphs.
