Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

Hajo Broersma, Fedor V. Fomin, Jan Kratochvil, Gerhard J. Woeginger
2005 Algorithmica  
2006) 'Planar graph coloring avoiding monochromatic subgraphs : trees and paths make it difficult.', Algorithmica., 44 (4). pp. 343-361. Further information on publisher's website: The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-profit purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata
more » ... cord in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract We consider the problem of coloring a planar graph with the minimum number of colors so 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 with at least one edge, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles. In particular, we prove that it is NP-complete to decide if a planar graph can be 2-colored so that no cycle of length at most 5 is monochromatic.
doi:10.1007/s00453-005-1176-8 fatcat:feseqsxyr5h5ddrp3kqsvk6bxu