On the Power of Color Refinement

  • V. Arvind
  • Johannes KöblerEmail author
  • Gaurav Rattan
  • Oleg Verbitsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9210)


Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color-refinement procedure succeeds in distinguishing G from any non-isomorphic graph H. Babai, Erdős, and Selkow (1982) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time \(O((n+m)\log n)\), where n and m denote the number of vertices and the number of edges in the input graph.


  1. 1.
    Arvind, V., Köbler, J., Rattan, G., Verbitsky, O.: On Tinhofer’s linear programming approach to isomorphism testing. In: In: Proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS), Lecture Notes in Computer Science. Springer (2015) (to appear)Google Scholar
  2. 2.
    Babai, L., Erdös, P., Selkow, S.M.: Random graph isomorphism. SIAM J. Comput. 9(3), 628–635 (1980)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Babai, L., Kucera, L.: Canonical labelling of graphs in linear average time. In: Proceedings of the 20th Annual Symposium on Foundations of Computer Science, pp. 39–46, (1979)Google Scholar
  4. 4.
    Berkholz, C., Bonsma, P., Grohe, M.: Tight lower and upper bounds for the complexity of canonical colour refinement. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 145–156. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  5. 5.
    Borri, A., Calamoneri, T., Petreschi, R.: Recognition of unigraphs through superposition of graphs. J. Graph Algorithms Appl. 15(3), 323–343 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Busacker, R., Saaty, T.: Finite Graphs and Networks: An Introduction with Applications. International Series in Pure and Applied Mathematics. McGraw-Hill Book Company, New York (1965)Google Scholar
  7. 7.
    Cardon, A., Crochemore, M.: Partitioning a graph in \(O(|A| \log _2 |V|)\). Theor. Comput. Sci. 19, 85–98 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Godsil, C.: Compact graphs and equitable partitions. Linear Algebra Appl. 255(13), 259–266 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Grohe, M., Kersting, K., Mladenov, M., Selman, E.: Dimension reduction via colour refinement. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 505–516. Springer, Heidelberg (2014) Google Scholar
  10. 10.
    Immerman, N., Lander, E.: Describing graphs: a first-order approach to graph canonization. In: Selman, A.L. (ed.) Complexity Theory Retrospective, pp. 59–81. Springer, Heidelberg (1990) CrossRefGoogle Scholar
  11. 11.
    Johnson, R.: Simple separable graphs. Pac. J. Math. 56, 143–158 (1975)CrossRefzbMATHGoogle Scholar
  12. 12.
    Kersting, K., Mladenov, M., Garnett, R., Grohe, M.: Power iterated color refinement. In: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 1904–1910. AAAI Press (2014)Google Scholar
  13. 13.
    Kiefer, S., Schweitzer, P., Selman, E.: Graphs identified by logics with counting. In:Graphs identified by logics with counting. In: Proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science (MFCS), Lecture Notes in Computer Science. Springer (2015) (to appear)Google Scholar
  14. 14.
    Koren, M.: Pairs of sequences with a unique realization by bipartite graphs. J. Comb. Theor. Series B 21(3), 224–234 (1976)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Krebs, A., Verbitsky, O.: Universal covers, color refinement, and two-variable logic with counting quantifiers: lower bounds for the depth. In: Proceedings of the 30-th ACM/IEEE Annual Symposium on Logic in Computer Science (LICS), IEEE Computer Society (2015) (to appear)Google Scholar
  16. 16.
    Ramana, M.V., Scheinerman, E.R., Ullman, D.: Fractional isomorphism of graphs. Discrete Math. 132(1–3), 247–265 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Shervashidze, N., Schweitzer, P., van Leeuwen, E.J., Mehlhorn, K., Borgwardt, K.M.: Weisfeiler-Lehman graph kernels. J. Mach. Learn. Res. 12, 2539–2561 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Tinhofer, G.: Graph isomorphism and theorems of Birkhoff type. Computing 36, 285–300 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Tyshkevich, R.: Decomposition of graphical sequences and unigraphs. Discrete Math. 220(1–3), 201–238 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Valiente, G.: Algorithms on Trees and Graphs. Springer, Heidelberg (2002) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • V. Arvind
    • 1
  • Johannes Köbler
    • 2
    Email author
  • Gaurav Rattan
    • 1
  • Oleg Verbitsky
    • 2
    • 3
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Institut für InformatikHumboldt Universität zu BerlinBerlinGermany
  3. 3.On leave from the Institute for Applied Problems of Mechanics and MathematicsLvivUkraine

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