Planar graph coloring with forbidden subgraphs : why trees and paths are dangerous

We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture for the case with a single forbidden connected (induced or non-induced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles