On the Number of Synchronizing Colorings of Digraphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9223)

Abstract

We deal with k-out-regular directed multigraphs with loops (called simply digraphs). The edges of such a digraph can be colored by elements of some fixed k-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring.

In the present paper we study how many synchronizing colorings can exist for a digraph with n vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to \(1-1/k^d\), for every \(d \ge 1\) and the number of vertices large enough.

On the basis of our results we state several conjectures and open problems. In particular, we conjecture that \(1-1/k\) is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for \(k=2\).

References

  1. 1.
    Adler, R.L., Goodwyn, L.W., Weiss, B.: Equivalence of topological Markov shifts. Israel J. Math. 27(1), 49–63 (1977)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ananichev, D.S., Volkov, M.V., Gusev, V.V.: Primitive digraphs with large exponents and slowly synchronizing automata. J. Math. Sci. 192(3), 263–278 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Béal, M.-P., Perrin, D.: A quadratic algorithm for road coloring. Discrete Appl. Math. 169, 15–29 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berlinkov, M.V.: On the probability of being synchronizable (2013). http://arxiv.org/abs/1304.5774
  5. 5.
    Cardoso, Â.: The Černý Conjecture and Other Synchronization Problems. Ph.D. thesis, University of Porto, Portugal (2014). http://hdl.handle.net/10216/73496
  6. 6.
    Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikálny Časopis Slovenskej Akadémie Vied 14(3), 208–216 (1964). In SlovakMATHGoogle Scholar
  7. 7.
    Cherubini, A.: Synchronizing and collapsing words. Milan J. Math. 75(1), 305–321 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eppstein, D.: Reset sequences for monotonic automata. SIAM J. Comput. 19, 500–510 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gusev, V.V., Pribavkina, E.V.: Reset thresholds of automata with two cycle lengths. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 200–210. Springer, Heidelberg (2014) Google Scholar
  10. 10.
    Jarvis, J.P., Shier, D.R.: Applied Mathematical Modeling: A Multidisciplinary Approach. Graph-theoretic analysis of finite Markov chains. CRC Press, Boca Raton (1996) Google Scholar
  11. 11.
    Kari, J., Volkov, M.V.: Černý’s conjecture and the road coloring problem. In: Handbook of Automata. European Science Foundation, to appearGoogle Scholar
  12. 12.
    Kisielewicz, A., Szykuła, M.: Generating small automata and the Černý conjecture. In: Konstantinidis, S. (ed.) CIAA 2013. LNCS, vol. 7982, pp. 340–348. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  13. 13.
    Kisielewicz, A., Szykuła, M.: Synchronizing Automata with Large Reset Lengths (2014). http://arxiv.org/abs/1404.3311
  14. 14.
    McKay, B.D., Piperno, A.: Practical graph isomorphism II. J. Symbolic Comput. 60, 94–112 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Nicaud, C.: Fast synchronization of random automata (2014). http://arxiv.org/abs/1404.6962
  16. 16.
    Roman, A.: P–NP threshold for synchronizing road coloring. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 480–489. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  17. 17.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Trahtman, A.N.: The road coloring problem. Isr. J. Math. 172(1), 51–60 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Volkov, M.V.: Open problems on synchronizing automata. Workshop “Around the Černý conjecture”, Wrocław (2008). http://csseminar.kadm.usu.ru/SLIDES/WroclawABCD2008/volkov_abcd_problems.pdf
  20. 20.
    Volkov, M.V.: Synchronizing automata and the černý conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  21. 21.
    Vorel, V., Roman, A.: Complexity of road coloring with prescribed reset words. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 161–172. Springer, Heidelberg (2015) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.ICTEAM InstituteUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Institute of Mathematics and Computer ScienceUral Federal UniversityEkaterinburgRussia
  3. 3.Institute of Computer ScienceUniversity of WrocławWrocławPoland

Personalised recommendations