Colorings with Few Colors: Counting, Enumeration and Combinatorial Bounds

  • Petr A. Golovach
  • Dieter Kratsch
  • Jean-Francois Couturier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)


We provide exact algorithms for enumeration and counting problems on edge colorings and total colorings of graphs, if the number of (available) colors is fixed and small. For edge 3-colorings the following is achieved: there is a branching algorithm to enumerate all edge 3-colorings of a connected cubic graph in time O *(25n/8). This implies that the maximum number of edge 3-colorings in an n-vertex connected cubic graph is O *(25n/8). Finally, the maximum number of edge 3-colorings in an n-vertex connected cubic graph is lower bounded by 12 n/10. Similar results are achieved for total 4-colorings of connected cubic graphs. We also present dynamic programming algorithms to count the number of edge k-colorings and total k-colorings for graphs of bounded pathwidth. These algorithms can be used to obtain fast exact exponential time algorithms for counting edge k-colorings and total k-colorings on graphs, if k is small.


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  1. 1.
    Beigel, R., Eppstein, D.: 3-coloring in time O(1.3289n). Journal of Algorithms 54, 168–204 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 575–582. IEEE, Los Alamitos (2006)CrossRefGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Narrow sieves for parameterized paths and packings, arXiv:1007.1161v1 (2010)Google Scholar
  4. 4.
    Bodlaender, H.L.: Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. J. Algorithms 11, 631–643 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eppstein, D.: Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 329–337. SIAM, Philadelphia (2001)Google Scholar
  6. 6.
    Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of lambda-labelings. Discrete Applied Mathematics 113, 59–72 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fomin, F.V., Gaspers, S., Saurabh, S.: Improved exact algorithms for counting 3- and 4-colorings. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 65–74. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Fomin, F.V., Gaspers, S., Saurabh, S.: On two techniques of combining branching and treewidth. Algorithmica 54, 181–207 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fomin, F., Grandoni, F., Kratsch, D.: Some new techniques in design and analysis of exact (exponential) algorithms. Bulletin of the EATCS 87, 47–77 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fomin, F.V., Grandoni, F., Pyatkin, A., Stepanov, A.: Combinatorial bounds via Measure and Conquer: Bounding minimal dominating sets and applications. ACM Transactions on Algorithms 5(1), Article 9 (2008)Google Scholar
  11. 11.
    Fomin, F.V., Høie, K.: Pathwidth of cubic graphs and exact algorithms. Inf. Process. Lett. 97, 191–196 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the Theory of NP-completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10, 718–720 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Koivisto, M.: An O(2n) Algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 2nd edn., pp. 583–590. IEEE, Los Alamitos (2006)CrossRefGoogle Scholar
  15. 15.
    Kowalik, L.: Improved edge-coloring with three colors. Theoret. Comp. Sci. 410, 3733–3742 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Král, D.: An exact algorithm for the channel assignment problem. Discrete Applied Mathematics 145, 326–331 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kratochvíl, J., Kratsch, D., Liedloff, M.: Exact algorithms for L(2,1)-labeling of graphs. In: Kucera, L., Kucera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 513–524. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rosenfeld, M.: On the total coloring of certain graphs. Israel Journal of Mathematics 9, 396–402 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sánchez-Arroyo, A.: Determining the total colouring number is NP-hard. Discrete Mathematics 78, 315–319 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Diskret. Anal., 25–30 (1964) (in Russian)Google Scholar
  22. 22.
    Zhou, X., Nishizeki, T.: Optimal parallel algorithm for edge-coloring partial k-trees with bounded degrees. In: Proceedings of the International Symposium on Parallel Architectures, Algorithms and Networks, pp. 167–174. IEEE, Los Alamitos (1994)CrossRefGoogle Scholar
  23. 23.
    Zhou, X., Nakano, S., Nishizeki, T.: Edge-coloring partial k-trees. J. Algorithms 21, 598–617 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Dieter Kratsch
    • 2
  • Jean-Francois Couturier
    • 2
  1. 1.School of Engineering and Computing SciencesDurham UniversityDurhamUnited Kingdom
  2. 2.Laboratoire d’Informatique Théorique et AppliquéeUniversité Paul Verlaine - MetzMetz Cedex 01France

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