Colouring Vertices of Triangle-Free Graphs

  • Konrad Dabrowski
  • Vadim Lozin
  • Rajiv Raman
  • Bernard Ries
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)


The vertex colouring problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it remains NP-complete even if we additionally exclude a graph F which is not a forest. We study the computational complexity of the problem in (K 3, F)-free graphs with F being a forest. From known results it follows that for any forest F on 5 vertices the vertex colouring problem is polynomial-time solvable in the class of (K 3, F)-free graphs. In the present paper, we show that the problem is also polynomial-time solvable in many classes of (K 3, F)-free graphs with F being a forest on 6 vertices.


Vertex colouring Triangle-free graphs Polynomial-time algorithm Clique-width 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balas, E., Yu, C.S.: On graphs with polynomially solvable maximum-weight clique problem. Networks 19, 247–253 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brandt, S.: Triangle-free graphs and forbidden subgraphs. Discrete Appl. Math. 120, 25–33 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brandt, S.: A 4-colour problem for dense triangle-free graphs. Discrete Math. 251, 33–46 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Klembt, T., Mahfud, S.: P 6- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Math. Theor. Comput. Sci. 8, 173–187 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Systems 33, 125–150 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Courcelle, B., Olariu, S.: Upper bounds to the clique-width of a graph. Discrete Applied Math. 101, 77–114 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dailey, D.P.: Uniqueness of colorability and colorability of planar 4 regular graphs are NP-complete. Discrete Math. 30, 289–293 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Diestel, R.: Graph theory, 3rd edn. Graduate Texts in Mathematics, vol. 173, pp. xvi+411. Springer, Berlin (2005)zbMATHGoogle Scholar
  10. 10.
    Hoang, C., Kaminski, M., Lozin, V., Sawada, J., Shu, X.: Deciding k-colorability of P 5-free graphs in polynomial time. Algorithmica 57, 74–81 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Computing 10, 718–720 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kamiński, M., Lozin, V.: Coloring edges and vertices of graphs without short or long cycles. Contributions to Discrete Mathematics 2, 61–66 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kamiński, M., Lozin, V.: Vertex 3-colorability of claw-free graphs. Algorithmic Operations Research 2, 15–21 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kamiński, M., Lozin, V.V., Milanič, M.: Recent developments on graphs of bounded clique-width. Discrete Applied Math. 157, 2747–2761 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kochol, M., Lozin, V., Randerath, B.: The 3-colorability problem on graphs with maximum degree 4. SIAM J. Computing 32, 1128–1139 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Král, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. In: Brandstädt, A., van Le, B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lozin, V.V.: Bipartite graphs without a skew star. Discrete Math. 257, 83–100 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lozin, V., Rautenbach, D.: On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM J. Discrete Math. 18, 195–206 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lozin, V., Volz, J.: The clique-width of bipartite graphs in monogenic classes. International Journal of Foundations of Computer Sci. 19, 477–494 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Maffray, F., Preissmann, M.: On the NP-completeness of the k-colorability problem for triangle-free graphs. Discrete Math. 162, 313–317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Randerath, B.: 3-colorability and forbidden subgraphs. I. Characterizing pairs. Discrete Math. 276, 313–325 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Randerath, B., Schiermeyer, I., Tewes, M.: Three-colourability and forbidden subgraphs. II. Polynomial algorithms. Discrete Math. 251, 137–153 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Randerath, B., Schiermeyer, I.: A note on Brooks’ theorem for triangle-free graphs. Australas. J. Combin. 26, 3–9 (2002)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Randerath, B., Schiermeyer, I.: Vertex colouring and forbidden subgraphs—a survey. Graphs Combin. 20, 1–40 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Randerath, B., Schiermeyer, I.: 3-colorability ∈ P for P 6-free graphs. Discrete Appl. Math. 136, 299–313 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Scott, A.D.: Induced trees in graphs of large chromatic number. J. Graph Theory 24, 297–311 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sgall, J., Wöginger, G.J.: The complexity of coloring graphs without long induced paths. Acta Cybernet. 15, 107–117 (2001)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Computing 6, 505–517 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Konrad Dabrowski
    • 1
  • Vadim Lozin
    • 1
  • Rajiv Raman
    • 1
  • Bernard Ries
    • 1
  1. 1.DIMAPUniversity of WarwickCoventryUK

Personalised recommendations