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Generation of Colourings and Distinguishing Colourings of Graphs

  • William Bird
  • Wendy Myrvold
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

A colouring of a graph \(X\) is an assignment of colours to the vertices of \(X\). A distinguishing colouring of \(X\) is a colouring such that no non-trivial automorphism of \(X\) preserves all colours. The distinguishing number of \(X\) is the minimum number of colours in a distinguishing colouring. This research presents a new algorithm for the generation of all colourings and all distinguishing colourings of a graph \(X\) up to isomorphism, and presents computational data on the distinguishing numbers of vertex transitive graphs.

Keywords

Graph distinguishability Combinatorial generation Graph theory 

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References

  1. 1.
    Albertson, M.O., Collins, K.L.: Symmetry breaking in graphs. Electronic Journal of Combinatorics 3(R18), 1–17 (1996)MathSciNetGoogle Scholar
  2. 2.
    Arvind, V., Cheng, C.T., Devanur, N.R.: On computing the distinguishing numbers of planar graphs and beyond: A counting approach. SIAM Journal on Discrete Mathematics 22(4), 1297–1324 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogstad, B., Cowen, L.: The distinguishing number of the hypercube. Discrete Mathematics 283(1), 29–35 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Butler, G. (ed.): Fundamental Algorithms for Permutation Groups. LNCS, vol. 559. Springer, Heidelberg (1991) zbMATHGoogle Scholar
  5. 5.
    Chan, M.: The maximum distinguishing number of a group. Electronic Journal of Combinatorics 13(R70), 1–8 (2006)Google Scholar
  6. 6.
    Chan, M.: The distinguishing number of the augmented cube and hypercube powers. Discrete Mathematics 308(11), 2330–2336 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheng, C.T.: On computing the distinguishing numbers of trees and forests. Electronic Journal of Combinatorics 13(R11), 1–12 (2006)Google Scholar
  8. 8.
    Furst, M., Hopcroft, J., Luks, E.: Polynomial-time algorithms for permutation groups. In: 21st Annual Symposium on Foundations of Computer Science, pp. 36–41, October 1980Google Scholar
  9. 9.
    Goldberg, L.: Automating Pólya theory: The computational complexity of the cycle index polynomial. Information and Computation 105(2), 268–288 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hall, M.: The Theory of Groups. Macmillan, New York (1959)zbMATHGoogle Scholar
  11. 11.
    Jerrum, M.: A compact representation for permutation groups. Journal of Algorithms 7(1), 60–78 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Knuth, D.E.: Efficient representation of perm groups. Combinatorica 11(1), 33–43 (1991)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Knuth, D.E.: The Art of Computer Programming, vol. 4A. Addison-Wesley, Reading (2011)Google Scholar
  14. 14.
    Kocay, W.: On writing isomorphism programs. Computational and Constructive Design Theory 368, 135–175 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    McKay, B.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)MathSciNetGoogle Scholar
  16. 16.
    Myrvold, W., Fowler, P.: Fast enumeration of all independent sets of a graph up to isomorphism. Journal of Combinatorial Mathematics and Combinatorial Computing 85, 173–194 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Potanka, K.S.: Groups, Graphs, and Symmetry-Breaking. Master’s thesis, Virginia Polytechnic Institute and State University (1998)Google Scholar
  18. 18.
    Royle, G.F., Praeger, C.E.: Constructing the vertex-transitive graphs of order 24. Journal of Symbolic Computation 8(4), 309–326 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Russell, A., Sundaram, R.: A note on the asymptotics and computational complexity of graph distinguishability. Electronic Journal of Combinatorics 5(R23), 1–7 (1998)MathSciNetGoogle Scholar
  20. 20.
    Sims, C.C.: Computation with permutation groups. In: Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation, pp. 23–28. ACM (1971)Google Scholar
  21. 21.
    Sims, C.C.: Computational methods in the study of permutation groups. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 169–183. Pergamon Press (1970)Google Scholar
  22. 22.
    Tymoczko, J.: Distinguishing numbers for graphs and groups. Electronic Journal of Combinatorics 11(R63), 1–13 (2004)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of VictoriaVictoriaCanada

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