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)


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.


Polynomial Time Chromatic Number Truth Assignment Free Graph Thick Edge 
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  1. 1.
    Balas, E., Yu, C.S.: On graphs with polynomially solvable maximum-weight clique problem. Networks 19, 247–253 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. In: Springer Graduate Texts in Mathematics, vol. 244 (2008)Google Scholar
  3. 3.
    Broersma, H.J., Fomin, F.V., Golovach, P.A., Paulusma, D.: Three complexity results on coloring P k-free graphs. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 95–104. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Broersma, H.J., Golovach, P.A., Paulusma, D., Song, J.: On coloring graphs without induced forests. In: ISAAC 2010 (to appear, 2010)Google Scholar
  5. 5.
    Bruce, D., Hoàng, C.T., Sawada, J.: A certifying algorithm for 3-colorability of P 5-free graphs. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 594–604. Springer, Heidelberg (2009)Google Scholar
  6. 6.
    Edwards, K.: The complexity of coloring problems on dense graphs. Theoret. Comput. Sci. 43, 337–343 (1986)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  8. 8.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hoàng, C.T., Kamiński, M., Lozin, V., Sawada, J., Shu, X.: Deciding k-colorability of P 5-free graphs in polynomial time. Algorithmica 57, 74–81 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Kratochvíl, J.: Precoloring extension with fixed color bound. Acta Math. Univ. Comen. 62, 139–153 (1993)MathSciNetMATHGoogle Scholar
  12. 12.
    Le, V.B., Randerath, B., Schiermeyer, I.: On the complexity of 4-coloring graphs without long induced paths. Theoret. Comput. Sci. 389, 330–335 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Randerath, B., Schiermeyer, I.: 3-Colorability ∈ P for P 6-free graphs. Discrete Appl. Math. 136, 299–313 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Randerath, B., Schiermeyer, I.: Vertex colouring and forbidden subgraphs - a survey. Graphs Combin. 20, 1–40 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Conference Record of the Tenth Annual ACM Symposium on Theory of Computing, San Diego, Calif., pp. 216–226. ACM, New York (1978)Google Scholar
  16. 16.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6, 505–517 (1977)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Tuza, Z.: Graph colorings with local restrictions - a survey. Discuss. Math. Graph Theory 17, 161–228 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Woeginger, G.J., Sgall, J.: The complexity of coloring graphs without long induced paths. Acta Cybernet. 15, 107–117 (2001)MathSciNetMATHGoogle Scholar

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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