Fragmented coloring of proper interval and split graphs

We study the fragmented coloring problem which is a generalization of the vertex coloring problem. A (,C)-fragmented coloring of a graph G is an assignment of a color from {0,,1} to each vertex of G such that every connected component in the graph induced by vertices of a single color has at most C vertices. For proper interval graphs, we give linear time and nearly linear time algorithms for the problems of minimizing given C, and minimizing C given . We also consider versions in which each vertex has a weight, and the sum of the weights in each induced connected component must be at most C. We give nearly linear time algorithms for the splittable weighted version and a 2-approximation for the non-splittable weighted version, all on proper interval graphs. We also prove that even the unweighted version is NP-hard for split graphs.