  Given a simple undirected graph $G=(V,E)$ and a partition of the vertex set
$V$ into $p$ parts, the \textsc{Partition Coloring Problem} asks if we can
select one vertex from each part of the partition such that the chromatic
number of the subgraph induced on the $p$ selected vertices is bounded by $k$.
PCP is a generalized problem of the classical \textsc{Vertex Coloring Problem}
and has applications in many areas, such as scheduling and encoding etc.
  In this paper, we show the complexity status of the \textsc{Partition
Coloring Problem} with three parameters: the number of colors, the number of
parts of the partition, and the maximum size of each part of the partition.
  Furthermore, we give a new exact algorithm for this problem.
