Exploring the complexity boundary between coloring and list-coloring

  • Flavia Bonomo
  • Guillermo Durán
  • Javier Marenco


Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the list-coloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs where the former admits polynomial-time algorithms but the latter is NP-complete. Our goal is to analyze the computational complexity of coloring problems lying “between” (from a computational complexity viewpoint) these two problems: precoloring extension, μ-coloring, and (γ,μ)-coloring.


Coloring Computational complexity List-coloring 


  1. Bandelt, H., & Mulder, H. (1986). Distance–hereditary graphs. Journal of Combinatorial Theory. Series B, 41, 182–208. CrossRefGoogle Scholar
  2. Bertossi, A. (1984). Dominating sets for split and bipartite graphs. Information Processing Letters, 19, 37–40. CrossRefGoogle Scholar
  3. Biro, M., Hujter, M., & Tuza, Zs. (1992). Precoloring extension. I. Interval graphs. Discrete Mathematics, 100(1–3), 267–279. CrossRefGoogle Scholar
  4. Bonomo, F., & Cecowski, M. (2005). Between coloring and list-coloring: μ-coloring. Electronic Notes in Discrete Mathematics, 19, 117–123. CrossRefGoogle Scholar
  5. Bonomo, F., Durán, G., & Marenco, J. (2006). Exploring the complexity boundary between coloring and list-coloring. Electronic Notes in Discrete Mathematics, 25, 41–47. CrossRefGoogle Scholar
  6. Booth, K., & Lueker, G. (1976). Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer Science and Technology, 13, 335–379. Google Scholar
  7. Brandstädt, A., Le, V., & Spinrad, J. (1999). Graph classes: A survey. Philadelphia: SIAM. Google Scholar
  8. Colbourn, C. J. (1984). The complexity of completing partial Latin squares. Annals of Discrete Mathematics, 8, 25–30. CrossRefGoogle Scholar
  9. Corneil, D., & Perl, Y. (1984). Clustering and domination in perfect graphs. Discrete Applied Mathematics, 9, 27–39. CrossRefGoogle Scholar
  10. Easton, T., Horton, S., & Parker, R. (2000). On the complexity of certain completion problems. Congressus Numerantium, 145, 9–31. Google Scholar
  11. Garey, M., Johnson, D., Miller, G., & Papadimitriou, C. (1980). The complexity of coloring circular arcs and chords. SIAM Journal on Algebraic and Discrete Methods, 1, 216–227. CrossRefGoogle Scholar
  12. Golumbic, M. (2004). Annals of discrete mathematics: Vol. 57. Algorithmic graph theory and perfect graphs (2nd ed.). Amsterdam: North–Holland. Google Scholar
  13. Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1, 169–197. CrossRefGoogle Scholar
  14. Hall, P. (1935). On representatives of subsets. Journal of the London Mathematical Society, 10, 26–30. CrossRefGoogle Scholar
  15. Hell, P. (2006). Personal communication. Google Scholar
  16. Holyer, I. (1981). The NP-completeness of edge-coloring. SIAM Journal on Computing, 10, 718–720. CrossRefGoogle Scholar
  17. Hujter, M., & Tuza, Zs. (1993). Precoloring extension. II. Graph classes related to bipartite graphs. Acta Mathematica Universitatis Comenianae, 62(1), 1–11. Google Scholar
  18. Hujter, M., & Tuza, Zs. (1996). Precoloring extension. III. Classes of perfect graphs. Combinatorics, Probability and Computing, 5, 35–56. CrossRefGoogle Scholar
  19. Jansen, K. (1997). The optimum cost chromatic partition problem. Lecture Notes in Computer Science, 1203, 25–36. Google Scholar
  20. Jansen, K., & Scheffler, P. (1997). Generalized coloring for tree-like graphs. Discrete Applied Mathematics, 75, 135–155. CrossRefGoogle Scholar
  21. König, D. (1916). Über graphen und ihre anwendung auf determinantentheorie und mengenlehre. Mathematische Annalen, 77, 453–465. CrossRefGoogle Scholar
  22. Kubale, M. (1992). Some results concerning the complexity of restricted colorings of graphs. Discrete Applied Mathematics, 36, 35–46. CrossRefGoogle Scholar
  23. Marx, D. (2006). Precoloring extension on unit interval graphs. Discrete Applied Mathematics, 154, 995–1002. CrossRefGoogle Scholar
  24. Nemhauser, G., & Wolsey, L. (1988). Wiley interscience series in discrete mathematics and optimization: Integer and combinatorial optimization. New York: Wiley. Google Scholar
  25. Tuza, Zs. (1997). Graph colorings with local constraints—a survey. Discussiones Mathematicae. Graph Theory, 17, 161–228. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Flavia Bonomo
    • 1
  • Guillermo Durán
    • 2
    • 3
  • Javier Marenco
    • 4
  1. 1.CONICET and Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Departamento de Ingeniería Industrial, Facultad de Ciencias Físicas y MatemáticasUniversidad de ChileSantiagoChile
  3. 3.CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  4. 4.Instituto de CienciasUniversidad Nacional de General SarmientoBuenos AiresArgentina

Personalised recommendations