, Volume 6, Issue 1–6, pp 869–891 | Cite as

Heuristics for rapidly four-coloring large planar graphs

  • Craig A. Morgenstern
  • Henry D. Shapiro


We present several algorithms for rapidly four-coloring large planar graphs and discuss the results of extensive experimentation with over 140 graphs from two distinct classes of randomly generated instances having up to 128,000 vertices. Although the algorithms can potentially require exponential time, the observed running times of our more sophisticated algorithms are linear in the number of vertices over the range of sizes tested. The use of Kempe chaining and backtracking together with a fast heuristic which usually, but not always, resolves impasses gives us hybrid algorithms that: (1) successfully four-color all our test graphs, and (2) in practice run, on average, only twice as slow as the well-known, nonexact, simple to code, Θ(n) saturation algorithm of Brélaz.

Key words

Four coloring Planar graphs Kempe chaining Saturation algorithm of Brélaz Heuristic algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. I. Appel, W. Haken, and J. Koch. 1977. Every Planar Map is Four Colorable. Part I: Discharging.Illinois J. Math.,21, 429–490.zbMATHMathSciNetGoogle Scholar
  2. [2]
    K. I. Appel, W. Haken, and J. Koch. 1977. Every Planar Map is Four Colorable. Part II: Reducibility.Illinois J. Math.,21, 491–567.zbMATHMathSciNetGoogle Scholar
  3. [3]
    R. Archuleta and H. Shapiro. 1986. A Fast Probabilistic Algorithm for Four-Coloring Large Planar Graphs.Proc. Fall 1986 Joint Computer Conference, pp. 595–600.Google Scholar
  4. [4]
    F. R. Bernhart. 1977. A Digest of the Four Color Theorem.J. Graph Theory,1, 207–225.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    D. Brélaz. 1979. New Methods to Color Vertices of a Graph.Comm. ACM.,22, 251–256.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    N. Chiba, T. Nishizeki, and N. Saito. 1981. A Linear 5-Coloring Algorithm of Planar Graphs.J. Algorithms,2, 317–327.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    F. D. J. Dunstan. 1976. Sequential Colourings of Graphs.Congr. Numer.,15, 151–158.MathSciNetGoogle Scholar
  8. [8]
    G. N. Frederickson. 1984. On Linear Time Algorithms for Five-Coloring Planar Graphs.Inform. Process. Lett.,19, 219–224.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    W. Haken. 1977. An Attempt to Understand the Four Color Problem.J. Graph Theory,1, 193–206.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. J. Heawood. 1890. Map-Colour Theorems.Quart. J. Math.,24, 332–338.Google Scholar
  11. [11]
    A. B. Kempe, 1879. On the Geographical Problem of the Four-Colors.Amer. J. Math. 2, 193–200.CrossRefMathSciNetGoogle Scholar
  12. [12]
    I. Kittell. 1935. A Group of Operations on a Partially Colored Map.Bull. Amer. Math. Soc.,41, 407–413.CrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Kubale and B. Jackowski. 1985. A Generalized Implicit Enumeration for Graph-Coloring.Comm. ACM.,28, 412–418.CrossRefGoogle Scholar
  14. [14]
    F. T. Leighton. 1979. A Graph Coloring Algorithm for Large Scheduling Problems.J. Res. Nat. Bur. Standards,84, 489–506.zbMATHMathSciNetGoogle Scholar
  15. [15]
    G. Marble and D. W. Matula. 1972. Computational Aspects of 4-Coloring Planar Graphs. Technical Report, University of Wisconsin.Google Scholar
  16. [16]
    D. W. Matula, G. Marble, and J. D. Isaacson. 1972. Graph Coloring Algorithms. InGraph Theory and Computing (R. C. Read, ed.), Academic Press, New York, pp. 109–122.Google Scholar
  17. [17]
    D. W. Matula, Y. Shiloach, and R. E. Tarjan. 1981. Analysis of Two Linear-Time Algorithms for Five-Coloring a Planar Graph.Congr. Numer.,33, 401.Google Scholar
  18. [18]
    C. Morgenstern. 1988. Saturation Based Graph Coloring Algorithms. Technical Report CS88-1, University of New Mexico, Albuquerque, New Mexico.Google Scholar
  19. [19]
    C. Morgenstern. 1990. Algorithms for General Graph Coloring. Doctoral Dissertation, Department of Computer Science, University of New Mexico, Albuquerque, New Mexico.Google Scholar
  20. [20]
    C. Morgenstern and H. D. Shapiro. 1984. Performance of Approximation Coloring Algorithms on Maximally Planar Graphs. Technical Report CS84-7, University of New Mexico, Albuquerque, New Mexico.Google Scholar
  21. [21]
    T. Nishizeki and N. Chiba. 1988. Planar Graphs: Theory and Algorithms.Ann. Discrete Math.,32.Google Scholar
  22. [22]
    J. Peemoller. 1983. A Correction to Brelaz's Modification of Brown's Coloring Algorithm.Comm. ACM,26, 595–597.CrossRefGoogle Scholar
  23. [23]
    T. L. Saaty and P. C. Kainen. 1986.The Four-Color Problem. Dover, New York.Google Scholar
  24. [24]
    M. H. Williams and K. T. Milne. 1984. The Performance of Algorithms for Colouring Planar Graphs.Comput. J.,27, 165–170.CrossRefGoogle Scholar
  25. [25]
    M. R. Williams. 1974. Heuristic Procedures.Software-Practice and Experience,4, 237–240.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Craig A. Morgenstern
    • 1
  • Henry D. Shapiro
    • 2
  1. 1.Department of Computer ScienceTexas Christian UniversityFort WorthUSA
  2. 2.Department of Computer ScienceUniversity of New MexicoAlbuquerqueUSA

Personalised recommendations