On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{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 $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

Lower bounds for the length of reset words in eulerian automata

For each odd n ≥ 5 we present a synchronizing Eulerian automaton with n states for which the minimum length of reset words is equal to n 2-3n+4/2. We also discuss various connections between the reset threshold of a synchronizing automaton and a sequence of reachability properties in its underlying graph. © 2013 World Scientific Publishing Company

An extensive English language bibliography on graph theory and its applications, supplement 1

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Lower Bounds for the Length of Reset words in Eulerian Automata

For each odd n ≥ 5 we present a synchronizing Eulerian automaton with n states for which the minimum length of reset words is equal to n 2-3n+4/2. We also discuss various connections between the reset threshold of a synchronizing automaton and a sequence of reachability properties in its underlying graph. © 2011 Springer-Verlag.Supported by the Russian Foundation for Basic Research, grant 10-01-00524, and by the Federal Education Agency of Russia, grant 2.1.1/13995