Planar Graph Coloring with Forbidden Subgraphs: Why Trees and Paths Are Dangerous

Authors:
Hajo Broersma;Fedor V. Fomin;Jan Kratochvíl;Gerhard J. Woeginger
Affiliations:
-;-;-;-
Venue:
SWAT '02 Proceedings of the 8th Scandinavian Workshop on Algorithm Theory
Year:
2002

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Abstract

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.