Abstract
A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented.
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References
R. L. Adler, L. W. Goodwyn and B. Weiss, Equivalence of topological Markov shifts, Israel Journal of Mathematics 27 (1977), 49–63.
R. L. Adler and B. Weiss, Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society, vol. 98, Providence, RI, 1970.
G. Budzban and A. Mukherjea, A semigroup approach to the Road Coloring Problem, Probability on Algebraic Structures. Contemporary Mathematics 261 (2000), 195–207.
A. Carbone, Cycles of relatively prime length and the road coloring problem, Israel Journal of Mathematics 123 (2001), 303–316.
K. Culik II, J. Karhumaki and J. Kari, A note on synchronized automata and Road Coloring Problem, in Developments in Language Theory (5th Int. Conf., Vienna, 2001), Lecture Notes in Computer Science, vol. 2295, 2002, pp. 175–185.
J. Friedman, On the road coloring problem, Proceedings of the American Mathematical Society 110 (1990), 1133–1135.
E. Gocka, W. Kirchherr, and E. Schmeichel, A note on the road-coloring conjecture, Ars Combinatoria. Charles Babbage Res. Centre, 49 (1998), 265–270.
R. Hegde and K. Jain, Min-Max theorem about the Road Coloring Conjecture, Euro-Comb 2005, DMTCS proc., AE, 2005, pp. 279–284.
N. Jonoska and S. Suen, Monocyclic decomposition of graphs and the road coloring problem, Congressum Numerantium 110 (1995), 201–209.
J. Kari, Synchronizing finite automata on Eulerian digraphs, in Mathematical foundations of computer science, 2001 (Mariánské Láznӗ), Lecture Notes in Comput. Sci., vol. 2136, Springer, Berlin, 2001, pp. 432–438.
D. Lind and B. Marcus, An Introduction of Symbolic Dynamics and Coding, Cambridge University Press, 1995.
A. Mateescu and A. Salomaa, Many-Valued Truth Functions, Černy’s conjecture and road coloring, Bulletin of the European Association for Theoretical Computer Science. EATCS 68 (1999), 134–148.
G. L. O’Brien, The road coloring problem, Israel Journal of Mathematics 39 (1981), 145–154.
D. Perrin and M. P. Schǔtzenberger, Synchronizing prefix codes and automata, and the road coloring problem, in Symbolic dynamics and its applications (New Haven, CT, 1991), Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 295–318.
J. E. Pin, On two combinatorial problems arising from automata theory, Annals of Discrete Math. 17 (1983), 535–548.
A. N. Trahtman, Notable trends concerning the synchronization of graphs and automata, in CTW2006—Cologne-Twente Workshop on Graphs and Combinatorial Optimization, Electron. Notes Discrete Math., vol. 25, Elsevier, Amsterdam, 2006, pp. 173–175.
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Trahtman, A.N. The road coloring problem. Isr. J. Math. 172, 51–60 (2009). https://doi.org/10.1007/s11856-009-0062-5
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DOI: https://doi.org/10.1007/s11856-009-0062-5