Skip to main content
SpringerLink
Log in

Colorings with few Colors: Counting, Enumeration and Combinatorial Bounds

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

Edge coloring, total coloring and L(2,1)-labeling are well-studied NP-hard graph problems. Even the versions asking whether a graph has a coloring with few colors or a labeling with few labels remain NP-hard on graphs of small maximum degree.

This paper studies enumeration and counting problems on edge colorings, total colorings and L(2,1)-labelings of graphs. One part deals with the enumeration of all edge 3-colorings, all total 4-colorings and all L(2,1)-labelings of span 5 of a given connected cubic graph. Branching algorithms to solve these enumeration problems are established. They imply upper bounds on the maximum number of edge 3-colorings, total 4-colorings and L(2,1)-labelings of span 5 in any n-vertex connected cubic graphs. Corresponding combinatorial lower bounds are also provided.

The other part of the paper studies dynamic programming algorithms solving counting problems. On one hand, algorithms to count the number of edge k-colorings and total k-colorings for graphs of bounded pathwidth are given. On the other hand, an algorithm to count the number of L(2,1)-labelings of span 4 for graphs of maximum degree three are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. For the purpose of this paper we shall address L(2,1)-labeling as a coloring problem.

  2. As has recently become standard, we write f(n)=O ∗(g(n)) if f(n)≤p(n)⋅g(n) for some polynomial p(n).

  3. Motivated by a presentation of the edge coloring results of our work, published in [18], in a talk given by D. Kratsch at a meeting of the AGAPE project in January 2011, S. Bessy and F. Havet first (during the meeting) established the maximum number of edge 3-colorings in cubic graphs, and then extended this in various directions [4]. Let us mention that their work improves upon our enumeration algorithms for edge 3-colorings and total 4-colorings of connected cubic graphs and the corresponding combinatorial upper bounds given in Sect. 3; they study neither the L(2,1)-labeling problem nor counting versions of the problems.

  4. The auxiliary graph H is the only multigraph of the paper and its only purpose is to ease the description of the example.

References

  1. Alon, N., Friedland, S.: The maximum number of perfect matchings in graphs with a given degree sequence. Electron. J. Comb. 15, N13 (2008)

    MathSciNet  Google Scholar 

  2. Alon, N., Rödl, V., Rucinski, A.: Perfect matchings in ϵ-regular graphs. Electron. J. Comb. 5, R13 (1998)

    Google Scholar 

  3. Beigel, R., Eppstein, D.: 3-coloring in time O(1.3289n). J. Algorithms 54, 168–204 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bessy, S., Havet, F.: Enumerating the edge-colourings and total colourings of a regular graph. J. Comb. Optim. (2012). doi:10.1007/s10878-011-9448-5

    Google Scholar 

  5. 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, New York (2006)

    Chapter  Google Scholar 

  6. Bodlaender, H.L.: Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. J. Algorithms 11, 631–643 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bregman, L.M.: Some properties of nonnegative matrices and their permanents. Sov. Math. Dokl. 14, 945–949 (1973)

    MATH  Google Scholar 

  8. 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 

  9. Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of lambda-labelings. Discrete Appl. Math. 113, 59–72 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fomin, F.V., Gaspers, S., Saurabh, S.: Improved exact algorithms for counting 3- and 4-colorings. In: COCOON 2007. Lecture Notes in Computer Science, vol. 4598, pp. 65–74. Springer, Berlin (2007)

    Google Scholar 

  11. Fomin, F.V., Gaspers, S., Saurabh, S.: On two techniques of combining branching and treewidth. Algorithmica 54, 181–207 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fomin, F., Grandoni, F., Kratsch, D.: Some new techniques in design and analysis of exact (exponential) algorithms. Bull. Eur. Assoc. Theor. Comput. Sci. 87, 47–77 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Fomin, F.V., Grandoni, F., Pyatkin, A., Stepanov, A.: Combinatorial bounds via measure and conquer: bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1), 9 (2008)

    Article  MathSciNet  Google Scholar 

  14. Fomin, F.V., Høie, K.: Pathwidth of cubic graphs and exact algorithms. Inf. Process. Lett. 97, 191–196 (2006)

    Article  MATH  Google Scholar 

  15. Fomin, F.V., Kratsch, D.: Exact exponential algorithms. In: Texts in Theoretical Computer Science. Springer, Berlin (2010)

    Google Scholar 

  16. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  17. Gaspers, S., Kratsch, D., Liedloff, M.: On independent sets and bicliques in graphs. Algorithmica 62, 637–658 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Golovach, P.A., Kratsch, D., Couturier, J.F.: Colorings with few colors: counting, enumeration and combinatorial bounds. In: Thilikos, D. (ed.) Proceedings of the 36th International Workshop on Graph Theoretic Concepts in Computer Science (WG 2010). Lecture Notes in Computer Science, vol. 6410, pp. 39–50. Springer, Berlin (2010)

    Google Scholar 

  19. Havet, F., Klazar, M., Kratochvíl, J., Kratsch, D., Liedloff, M.: Exact algorithms for L(2,1)-labeling of graphs. Algorithmica 59, 169–194 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10, 718–720 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Junosza-Szaniawski, K., Kratochvíl, J., Liedloff, M., Rossmanith, P., Rzazewski, P.: Fast exact algorithm for L(2,1)-labeling of graphs. In: Ogihara, M., Tarui, J. (eds.) Proceedings of the 8th Annual Conference on Theory and Applications of Models of Computation (TAMC 2011). Lecture Notes in Computer Science, vol. 6648, pp. 82–93. Springer, Berlin (2011)

    Chapter  Google Scholar 

  22. Junosza-Szaniawski, K., Rzazewski, P.: On improved exact algorithms for L(2,1)-labeling of graphs. In: Iliopoulos, C.S., Smyth, W.F. (eds.) Proceedings of the 21st International Workshop on Combinatorial Algorithms (IWOCA 2010). Lecture Notes in Computer Science, vol. 6460, pp. 34–37. Springer, Berlin (2010)

    Chapter  Google Scholar 

  23. Junosza-Szaniawski, K., Rzazewski, P.: On the complexity of exact algorithm for L(2,1)-labeling of graphs. Inf. Process. Lett. 111, 697–701 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kloks, T.: Treewidth, Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  25. 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), pp. 583–590. IEEE, New York (2006)

    Chapter  Google Scholar 

  26. Kowalik, L.: Improved edge-coloring with three colors. Theor. Comput. Sci. 410, 3733–3742 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Král, D.: An exact algorithm for the channel assignment problem. Discrete Appl. Math. 145, 326–331 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Moon, J.W., Moser, L.: On cliques in graphs. Isr. J. Math. 3, 23–28 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rosenfeld, M.: On the total coloring of certain graphs. Isr. J. Math. 9, 396–402 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sánchez-Arroyo, A.: Determining the total colouring number is NP-hard. Discrete Math. 78, 315–319 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. Szegedy, C.: On the number of 3-edge colorings of cubic graphs. Eur. J. Comb. 23, 113–120 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Diskretn. Anal. 3, 25–30 (1964) (in Russian)

    MathSciNet  Google Scholar 

  33. Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Combinatorial Optimization—Eureka, You Shrink! Lecture Notes in Computer Science, vol. 2570, pp. 185–207. Springer, Berlin (2003)

    Chapter  Google Scholar 

  34. 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, New York (1994)

    Chapter  Google Scholar 

  35. Zhou, X., Nakano, S., Nishizeki, T.: Edge-coloring partial k-trees. J. Algorithms 21, 598–617 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

We would like to thank anonymous referees for their careful reading of an earlier version and the helpful comments.

Author information

Authors and Affiliations

  1. Laboratoire d’Informatique Théorique et Appliquée, Université Paul Verlaine – Metz, 57045, Metz Cedex 01, France

    Jean-Francois Couturier & Dieter Kratsch

  2. School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK

    Petr A. Golovach & Artem Pyatkin

  3. Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, 45067, Orléans Cedex 2, France

    Mathieu Liedloff

Authors
  1. Jean-Francois Couturier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr A. Golovach
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dieter Kratsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mathieu Liedloff
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Artem Pyatkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dieter Kratsch.

Additional information

An extended abstract of this paper was presented at the 36th International Workshop on Graph Theoretic Concepts in Computer Science, Zarós, Crete, Greece, June 28–30, 2010 and published in the Proceedings of WG 2010 [18].

J.-F. Couturier, D. Kratsch and M. Liedloff were supported by ANR under project AGAPE (ANR-09-BLAN-0159-03).

P.A. Golovach was supported by EPSRC under project EP/G043434/1.

A. Pyatkin was supported by EPSRC Grant EP/F064551/1, by the RFBR projects 12-01-00184-a and 12-01-00093-a, and by the Ministry of education and science of the Russian Federation (contract number 14.740.11.0868).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Couturier, JF., Golovach, P.A., Kratsch, D. et al. Colorings with few Colors: Counting, Enumeration and Combinatorial Bounds. Theory Comput Syst 52, 645–667 (2013). https://doi.org/10.1007/s00224-012-9410-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-012-9410-7

Keywords

Search

Navigation