Component Coloring of Proper Interval Graphs and Split Graphs

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