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.
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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.
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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
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