A Partially Synchronizing Coloring

  • Avraham N. Trahtman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6072)


Given a finite directed graph, a coloring of its edges turns the graph into a finite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected finite graph of a uniform outdegree (constant outdegree of any vertex) is k-synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a k-synchronizing word.

For k = 1 one has the well known road coloring problem. The recent positive solution of the road coloring problem implies an elegant generalization considered first by Béal and Perrin: a directed finite strongly connected graph of uniform outdegree is k-synchronizing iff the greatest common divisor of lengths of all its cycles is k.

Some consequences for coloring of an arbitrary finite digraph are presented. We describe a subquadratic algorithm of the road coloring for the k-synchronization implemented in the package TESTAS. A new linear visualization program demonstrates the obtained coloring. Some consequences for coloring of an arbitrary finite digraph and of such a graph of uniform outdegree are presented.


graph algorithm synchronization road coloring deterministic finite automaton 


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  1. 1.
    Adler, R.L., Weiss, B.: Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc., Providence, RI, 98 (1970)Google Scholar
  2. 2.
    Aho, A., Hopcroft, J., Ulman, J.: The Design and Analisys of Computer Algorithms. Addison-Wesley, Reading (1974)Google Scholar
  3. 3.
    Bauer, T., Cohen, N., Trahtman, A.N.: The visualization of digraph based on its structure properties. In: RTAGS, Ramat-Gan, Israel (2007)Google Scholar
  4. 4.
    Béal, M.P., Perrin, D.: A quadratic algorithm for road coloring (2008), arXiv: 0803.0726v2 [cs.DM]Google Scholar
  5. 5.
    Budzban, G., Feinsilver, P.: The Generalized Road Coloring Problem (2008), arXiv:0903.0192v, [cs.DM]Google Scholar
  6. 6.
    Culik, K., Karhumaki, J., Kari, J.: A note on synchronized automata and Road Coloring Problem. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 175–185. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Černy, J.: Poznamka k homogenym eksperimentom s konechnymi automatami. Math.-Fyz. Čas. 14, 208–215 (1964)zbMATHGoogle Scholar
  8. 8.
    Kari, J.: Synchronizing finite automata on Eulerian digraphs. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 432–438. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Mateescu, A., Salomaa, A.: Many-valued truth function. Černy conjuncture and road coloring, EATCS Bulletin 68, 134–150 (1999)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Trahtman, A.N.: Synchronizing Road Coloring. In: 5-th IFIP WCC-TCS, vol. 273, pp. 43–53. Springer, Heidelberg (2008)Google Scholar
  11. 11.
    Trahtman, A.N.: The Road Coloring and Cerny Conjecture. In: Proc. of Prague Stringology Conference, pp. 1–12 (2008)Google Scholar
  12. 12.
    Trahtman, A.N.: The Road Coloring for Mapping on k States (2008), arXiv:0812.4798, [cs.DM]Google Scholar
  13. 13.
    Trahtman, A.N.: The road coloring problem. Israel Journal of Math. 1(172), 51–60 (2009)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Avraham N. Trahtman
    • 1
  1. 1.Dep. of Math.Bar-Ilan UniversityRamat GanIsrael

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