  We introduce a generalization of the well known graph (vertex) coloring
problem, which we call the problem of \emph{component coloring of graphs}.
Given a graph, the problem is to color the vertices using minimum number of
colors so that the size of each connected component of the subgraph induced by
the vertices of the same color does not exceed $C$. We give a linear time
algorithm for the problem on proper interval graphs. We extend this algorithm
to solve two weighted versions of the problem in which vertices have integer
weights. In the \emph{splittable} version the weights of vertices can be split
into differently colored parts, however, the total weight of a monochromatic
component cannot exceed $C$. For this problem on proper interval graphs we give
a polynomial time algorithm. In the \emph{non-splittable} version the vertices
cannot be split. Using the algorithm for the splittable version we give a
2-approximation algorithm for the non-splittable problem on proper interval
graphs which is NP-hard. We also prove that even the unweighted version of the
problem is NP-hard for split graphs.
