Abstract
The 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, finite automata, coding and symbolic dynamics. Many partial solutions of the problem have been found and different generalizations were considered.
The positive solution of the road coloring problem is presented below. We reproduce from the literature also the statements used in our proof. The necessary and sufficient conditions of synchronizing road coloring of directed graph with constant outdegree of a vertex are presented.
Chapter PDF
Similar content being viewed by others
References
R.L. Adler, L.W. Goodwyn, B. Weiss. Equivalence of topological Markov shifts, Israel J. of Math. 27, 49-63, 1977.
R.L. Adler, B. Weiss. Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc. 98, Providence, RI, 1970.
G. Budzban, A. Mukherjea. A semigroup approach to the Road Coloring Problem, Probability on Algebraic Structures. Contemporary Mathematics, 261, 195-207, 2000.
A. Carbone. Cycles of relatively prime length and the road coloring problem, Israel J. of Math., 123, 303-316, 2001.
K. Culik II, J. Karhumaki, J. Kari. A note on synchronized automata and Road Coloring Problem, Developments in Language Theory (5th Int. Conf., Vienna, 2001), Lecture Notes in Computer Science, 2295, 175-185, 2002.
J. Friedman. On the road coloring problem, Proc. of the Amer. Math. Soc. 110, 1133-1135, 1990.
E. Gocka, W. Kirchherr, E. Schmeichel, A note on the road-coloring conjecture. Ars Combin. 49, 265-270, 1998.
R. Hegde, K. Jain, Min-Max theorem about the Road Coloring Conjecture EuroComb 2005, DMTCS proc., AE, 279-284, 2005.
N. Jonoska, S. Suen. Monocyclic decomposition of graphs and the road coloring problem, Congressum numerantium, 110, 201-209, 1995.
J. Kari. Synchronizing finite automata on Eulerian digraphs, Springer, Lect. Notes in Comp. Sci., 2136, 432-438, 2001.
P. Lankaster. Theory of Matrices, Acad. Press, NY - London, 1969.
D. Lind, B. Marcus. An Introduction of Symbolic Dynamics and Coding, Cambridge Univ. Press, 1995.
A. Mateescu, A. Salomaa, Many-Valued Truth Functions, Černy’s Conjecture and Road Coloring, Bull. of European Ass. for TCS, 68, 134-148, 1999.
G.L. O’Brien. The road coloring problem, Israel J. of Math., 39, 145-154, 1981.
D. Perrin, M.P. Schŭtzenberger. Synchronizing prefix codes and automata, and the road coloring problem, In Symbolic Dynamics and Appl., Contemp. Math., 135, 295-318, 1992.
J.E. Pin. On two combinatorial problems arising from automata theory, Annals of Discrete Math., 17, 535-548, 1983.
A.N. Trahtman. Notable trends concerning the synchronization of graphs and automata, CTW06, El. Notes in Discrete Math., 25, 173-175, 2006.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 IFIP International Federation for Information Processing
About this paper
Cite this paper
Trahtman, A. (2008). Synchronizing Road Coloring. In: Ausiello, G., Karhumäki, J., Mauri, G., Ong, L. (eds) Fifth Ifip International Conference On Theoretical Computer Science – Tcs 2008. IFIP International Federation for Information Processing, vol 273. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09680-3_3
Download citation
DOI: https://doi.org/10.1007/978-0-387-09680-3_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-09679-7
Online ISBN: 978-0-387-09680-3
eBook Packages: Computer ScienceComputer Science (R0)