Narrowing Down the Gap on the Complexity of Coloring Pk-Free Graphs

  • Hajo Broersma
  • Petr A. Golovach
  • Daniël Paulusma
  • Jian Song
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

A graph is P k -free if it does not contain an induced subgraph isomorphic to a path on k vertices. We show that deciding whether a P 8-free graph can be colored with at most four colors is an NP-complete problem. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-coloring is NP-complete for P 9-free graphs, and a result of Woeginger and Sgall, who showed that 5-coloring is NP-complete for P 8-free graphs. Additionally, we prove that the pre-coloring extension version of 4-coloring is NP-complete for P 7-free graphs, but that the pre-coloring extension version of 3-coloring is polynomially solvable for (P 2 + P 4)-free graphs, a subclass of P 7-free graphs.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hajo Broersma
    • 1
  • Petr A. Golovach
    • 1
  • Daniël Paulusma
    • 1
  • Jian Song
    • 1
  1. 1.School of Engineering and Computing Sciences, Science LaboratoriesDurham UniversityDurhamUK

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