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

Haitze J. Broersma, F.V. Fomin, J. Kratochvil, Gerhard Woeginger

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14 Citations (Scopus)

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 noninduced) 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.
Original languageUndefined
Article number10.1007/s00453-005-1176-8
Pages (from-to)343-361
Number of pages19
JournalAlgorithmica
Volume44
Issue number4
DOIs
Publication statusPublished - May 2006

Keywords

  • METIS-228211
  • EWI-8089
  • IR-63664

Cite this

Broersma, H. J., Fomin, F. V., Kratochvil, J., & Woeginger, G. (2006). Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult. Algorithmica, 44(4), 343-361. [10.1007/s00453-005-1176-8]. https://doi.org/10.1007/s00453-005-1176-8