On the complexity of the selective graph coloring problem in some special classes of graphs

In this paper, we consider the selective graph coloring problem. Given an integer k ≥ 1 and a graph G = (V, E) with a partition V1, ..., Vp of V, it consists in deciding whether there exists a set V* in G such that | V* ∩ Vi | = 1 for all i ∈ {1, ..., p}, and such that the graph induced by V* is k-colorable. We investigate the complexity status of this problem in various classes of graphs.