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 \(k=2\).
V.V. Gusev—Supported by the Communauté française de Belgique - Actions de Recherche Concertées, by the Belgian Programme on Interuniversity Attraction Poles, by the Russian foundation for basic research (grant 13-01-00852), Ministry of Education and Science of the Russian Federation (project no. 1.1999.2014/K), Presidential Program for Young Researchers (grant MK-3160.2014.1) and the Competitiveness Program of Ural Federal University.
M. Szykuła—Supported in part by Polish NCN grant DEC-2013/09/N/ST6/01194.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adler, R.L., Goodwyn, L.W., Weiss, B.: Equivalence of topological Markov shifts. Israel J. Math. 27(1), 49–63 (1977)
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)
Béal, M.-P., Perrin, D.: A quadratic algorithm for road coloring. Discrete Appl. Math. 169, 15–29 (2014)
Berlinkov, M.V.: On the probability of being synchronizable (2013). http://arxiv.org/abs/1304.5774
Cardoso, Â.: The Černý Conjecture and Other Synchronization Problems. Ph.D. thesis, University of Porto, Portugal (2014). http://hdl.handle.net/10216/73496
Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikálny Časopis Slovenskej Akadémie Vied 14(3), 208–216 (1964). In Slovak
Cherubini, A.: Synchronizing and collapsing words. Milan J. Math. 75(1), 305–321 (2007)
Eppstein, D.: Reset sequences for monotonic automata. SIAM J. Comput. 19, 500–510 (1990)
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)
Jarvis, J.P., Shier, D.R.: Applied Mathematical Modeling: A Multidisciplinary Approach. Graph-theoretic analysis of finite Markov chains. CRC Press, Boca Raton (1996)
Kari, J., Volkov, M.V.: Černý’s conjecture and the road coloring problem. In: Handbook of Automata. European Science Foundation, to appear
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)
Kisielewicz, A., Szykuła, M.: Synchronizing Automata with Large Reset Lengths (2014). http://arxiv.org/abs/1404.3311
McKay, B.D., Piperno, A.: Practical graph isomorphism II. J. Symbolic Comput. 60, 94–112 (2014)
Nicaud, C.: Fast synchronization of random automata (2014). http://arxiv.org/abs/1404.6962
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)
Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)
Trahtman, A.N.: The road coloring problem. Isr. J. Math. 172(1), 51–60 (2009)
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
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)
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)
Acknowledgment
The authors want to thank Mikhail Volkov for his significant contributions to the theory of synchronizing automata on the occasion of his \(60^{\text {th}}\) birthday.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Gusev, V.V., Szykuła, M. (2015). On the Number of Synchronizing Colorings of Digraphs. In: Drewes, F. (eds) Implementation and Application of Automata. CIAA 2015. Lecture Notes in Computer Science(), vol 9223. Springer, Cham. https://doi.org/10.1007/978-3-319-22360-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-22360-5_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22359-9
Online ISBN: 978-3-319-22360-5
eBook Packages: Computer ScienceComputer Science (R0)