Advertisement

Open Problems on Graph Coloring for Special Graph Classes

  • Daniël PaulusmaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping \(c: V\rightarrow \{1,2,\ldots \}\) such that \(c(u)\ne c(v)\) for every edge \(uv\in E\). We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring.

References

  1. 1.
    Aboulker, P., Brettell, N., Havet, F., Marx, D., Trotignon, N.: Colouring graphs with constraints on connectivity, Manuscript. arXiv:1505.01616
  2. 2.
    Appel, K., Haken, W.: Every planar map is four colorable. In: Contemporary Mathematics, vol. 89. AMS Bookstore (1989)Google Scholar
  3. 3.
    Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23, 11–24 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonomo, F., Durán, G., Marenco, J.: Exploring the complexity boundary between coloring and list-coloring. Ann. Oper. Res. 169, 3–16 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Klembt, T., Mahfud, S.: \(P_6\)- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Math. Theor. Comput. Sci. 8, 173–188 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Broersma, H.J., Capponi, A., Paulusma, D.: A new algorithm for on-line coloring bipartite graphs. SIAM J. Discrete Math. 22, 72–91 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Broersma, H.J., Kloks, T., Kratsch, D., Müller, H.: Independent sets in asteroidal triple-free graphs. SIAM J. Discrete Math. 12, 276–287 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brooks, R.L.: On colouring the nodes of a network. Math. Proc. Cambridge Philos. Soc. 37, 194–197 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cai, L.: Parameterized complexity of vertex coloring. Discrete Appl. Math. 127, 415–429 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cao, Y.: Linear recognition of almost (unit) interval graphs, Manuscript. arXiv:1403.1515
  11. 11.
    Cieślik, I., Kozik, M., Micek, P.: On-line coloring of \(I_s\)-free graphs and co-planar graphs. Discrete Math. Theor. Comput. Sci. Proc. AF, 61–68 (2006)zbMATHGoogle Scholar
  12. 12.
    Chlebík, M., Chlebíková, J.: Hard coloring problems in low degree planar bipartite graphs. Discrete Appl. Math. 154, 1960–1965 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chudnovsky, M.: Coloring graphs with forbidden induced subgraphs. Proc. ICM IV, 291–302 (2014)Google Scholar
  14. 14.
    Chudnovsky, M., Robertson, N., Seymour, P.D., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Corneil, D.G., Rotics, U.: On the relationship between clique-width and treewidth. SIAM J. Comput. 34, 825–847 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dabrowski, K.K., Dross, F., Johnson, M., Paulusma, D.: Filling the complexity gaps for colouring planar, bounded degree graphs, Manuscript. arXiv:1506.06564
  17. 17.
    Demange, M., de Werra, D.: On some coloring problems in grids. Theoret. Comput. Sci. 472, 9–27 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Doucha, M., Kratochvíl, J.: Cluster vertex deletion: a parameterization between vertex cover and clique-width. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 348–359. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Emden-Weinert, T., Hougardy, S., Kreuter, B.: Uniquely colourable graphs and the hardness of colouring graphs of large girth. Comb. Probab. Comput. 7, 375–386 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Enright, J., Stewart, L., Tardos, G.: On list coloring and list homomorphism of permutation and interval graphs. SIAM J. Discrete Math. 28, 1675–1685 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fellows, M.R., Fomin, F.V., Lokshtanov, D., Rosamond, F., Saurabh, S., Szeider, S., Thomassen, C.: On the complexity of some colorful problems parameterized by treewidth. Inf. Comput. 209, 143–153 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Felsner, S., Micek, P., Ueckerdt, T.: On-line coloring between two lines. In: Proceedings SoCG 2015, LIPIcs, vol. 34, pp. 630–641 (2015)Google Scholar
  23. 23.
    Fiala, J., Golovach, P.A., Kratochvíl, J.: Parameterized complexity of coloring problems: treewidth versus vertex cover. Theoret. Comput. Sci. 412, 2514–2523 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Clique-width: on the price of generality. In: Proceedings of SODA 2009, pp. 825–834 (2009)Google Scholar
  25. 25.
    Ganian, R.: Twin-cover: beyond vertex cover in parameterized algorithmics. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 259–271. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  26. 26.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. In: Proceedings of STOC, pp. 47–63 (1974)Google Scholar
  27. 27.
    Golovach, P.A., Johnson, M., Paulusma, D., Song, J.: A survey on the computational complexity of coloring graphs with forbidden subgraphs, Manuscript. arXiv:1407.1482
  28. 28.
    Golovach, P.A., Paulusma, D., Song, J.: Closing complexity gaps for coloring problems on H-free graphs. Inf. Comput. 237, 20–21 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. Int. J. Found. Computer Sci. 11, 423–443 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. Ann. Discret. Math. 21, 325–356 (1984)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: distance from triviality. IWPEC 2004. LNCS, vol. 3162, pp. 162–173. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  32. 32.
    Gyárfás, A., Király, Z., Lehel, J.: On-line competitive coloring algorithms. Technical report TR-9703-1 (1997)Google Scholar
  33. 33.
    Gyárfás, A., Lehel, J.: Effective on-line coloring of \(P_5\)-free graphs. Combinatorica 11, 181–184 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gyárfás, A., Lehel, J.: First fit and on-line chromatic number of families of graphs. Ars Combinatorica 29C, 168–176 (1990)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Gyárfás, A., Lehel, J.: On-line and first-fit colorings of graphs. J. Graph Theory 12, 217–227 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Golovach, P.A., Paulusma, D.: List coloring in the absence of two subgraphs. Discrete Appl. Math. 166, 123–130 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford Univ, Press (2004)CrossRefzbMATHGoogle Scholar
  38. 38.
    Hell, P., Nešetřil, J.: On the complexity of \(H\)-coloring. J. Comb. Theory Ser. B 48, 92–110 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Huang, S., Johnson, M., Paulusma, D.: Narrowing the complexity gap for coloring \((C_s, P_t)\)-Free Graphs. Comput. J. (to appear)Google Scholar
  40. 40.
    Hujter, M., Tuza, Z.: Precoloring extension. II. Graph classes related to bipartite graphs. Acta Math. Univ. Comenianae LXII, 1–11 (1993)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Hujter, M., Tuza, Z.: Precoloring extension. III. Classes of perfect graphs. Comb. Probab. Comput. 5, 35–56 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Jansen, K.: Complexity results for the optimum cost chromatic partition problem. Universität Trier, Mathematik/Informatik, Forschungsbericht, pp. 96–41 (1996)Google Scholar
  43. 43.
    Jansen, B.M.P., Kratsch, S.: Data reduction for graph coloring problems. FCT 2011. LNCS, vol. 6914, pp. 90–101. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  44. 44.
    Jansen, K., Scheffler, P.: Generalized coloring for tree-like graphs. Discrete Appl. Math. 75, 135–155 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kierstead, H.A.: Coloring graphs on-line. In: Fiat, A. (ed.) Online Algorithms 1996. LNCS, vol. 1442, pp. 281–305. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  46. 46.
    Kierstead, H.A.: The linearity of first-fit coloring of interval graphs. SIAM J. Discrete Math. 1, 526–530 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Kierstead, H.A., Penrice, S.G., Trotter, W.T.: On-line coloring and recursive graph theory. SIAM J. Discrete Math. 7, 72–89 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Appl. Math. 126, 197–221 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. WG 2001. LNCS, vol. 2204, p. 254. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  50. 50.
    Kratochvíl, J.: Precoloring extension with fixed color bound. Acta Mathematica Universitatis Comenianae 62, 139–153 (1993)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Kratochvíl, J., Tsuza, Z.: Algorithmic complexity of list colorings. Discrete Appl. Math. 50, 297–302 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Lovász, L.: Coverings and coloring of hypergraphs. In: Proceedings of 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Math, pp. 3–12 (1973)Google Scholar
  53. 53.
    Kratsch, D., Müller, H.: Colouring AT-free graphs. ESA 2012. LNCS, vol. 7501, pp. 707–718. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  54. 54.
    Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57, 747–768 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Marx, D.: Parameterized coloring problems on chordal graphs. Theoret. Comput. Sci. 351, 407–424 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Marx, D.: Precoloring extension on unit interval graphs. Discrete Appl. Math. 154, 995–1002 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Mertzios, G.B., Spirakis, P.G.: Algorithms and almost tight results for 3-colorability of small diameter graphs, Algorithmica (to appear)Google Scholar
  58. 58.
    Micek, P., Wiechert, V.: An on-line competitive algorithm for coloring P8-free bipartite graphs. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 516–527. Springer, Heidelberg (2014)Google Scholar
  59. 59.
    Micek, P., Wiechert, V.: An on-line competitive algorithm for coloring bipartite graphs without long induced paths, Manuscript. arXiv:1502.00859
  60. 60.
    Molloy, M., Reed, B.: Colouring graphs when the number of colours is almost the maximum degree. J. Comb. Theory, Ser. B 109, 134–195 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Randerath, B., Schiermeyer, I.: Vertex coloring and forbidden subgraphs - a survey. Graphs Comb. 20, 1–40 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Oum, S.-L., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory Ser. B 96, 514–528 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Stacho, J.: 3-coloring AT-free graphs in polynomial time. Algorithmica 64, 384–399 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Takenaga, Y., Higashide, K.: Vertex coloring of comparability+ke and –ke graphs. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 102–112. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  65. 65.
    Toft, B., Jensen, J.R.: Graph Coloring Problems. Wiley, New York (1995)zbMATHGoogle Scholar
  66. 66.
    Villanger, Y., Heggernes, P., Paul, C., Telle, J.A.: Interval completion is fixed parameter tractable. SIAM J. Comput. 38, 2007–2020 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Vizing, V.G.: Coloring the vertices of a graph in prescribed colors. In: Diskret. Analiz., no. 29, Metody Diskret. Anal. v. Teorii Kodov i Shem, vol. 101, pp. 3–10 (1976)Google Scholar
  68. 68.
    Vušković, K.: Even-hole-free graphs: a survey. Appl. Anal. Discrete Math. 4, 219–240 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    de Weijer, P.: Kernelization upper bounds for parameterized graph coloring problems. MSc Thesis, Utrecht University (2013)Google Scholar
  70. 70.
    Tuza, Z.: Graph colorings with local restrictions - a survey. Discussiones Mathematicae Graph Theory 17, 161–228 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Engineering and Computing SciencesDurham University Science LaboratoriesDurhamUK

Personalised recommendations