1. Road coloring Complete deterministic directed finite automaton with transition graph Γ For edge q → p suppose p= q ά. For a set of states Q and mapping ά consider a map Qά and Qs for s=ά 1 ά 2 … ά i. Γs presents a map of Γ. Synchronizing coloring of directed graph Γ turns the graph into a synchronizing automaton If for some word s |Γs|=1 then s is synchronizing word of automaton with transition graph Γ and the automaton is called synchronizing. Deterministic a a Complete – for any vertex outgoing edges of all colors from given alphabet ά q ά qάqά

2. Adler, Goodwyn, Weiss, 1970, 1977, Road coloring problem AGW graph 1.directed finite strongly connected graph 2.constant outdegree of all its vertices 3. the greatest common divisor (gcd) of lengths of all its cycles is one. Has AGW graph synchronizing coloring? The problem awaked unusual interest the problem depends only on sink (minimal) strongly connected component constant outdegree - for to be complete and deterministic the condition on gcd is necessary

3. Known T 1. (Kari, 2001, Culik, Karhumaki, Kari, 2000) Let us consider a coloring of AGW graph Γ. Let ρ be binary relation on the set of states. For any stable pair of states p, q suppose p ρ q. Then ρ is a congruence relation, Γ /ρ presents an AGW graph and synchronizing coloring of Γ /ρ implies synchronizing recoloring of Γ. A pair of distinct states p, q is called synchronizing if ps = qs for some s € Σ +. In opposite case, if for any s ps ≠ qs we call the pair deadlock. (Last relation is stable) Synchronizing pair p, q is called stable if for any s the pair (ps, qs) is also synchronizing. (Culik, Karhumaki, Kari)

4. L.2 Let F be F-clique via some coloring of AGW graph Γ. For any word s the set Fs is also an F-clique and any state p belongs to some F-clique. L.3 Let A and B (|A|>1) be F-cliques via some coloring of the AGW graph Γ and |A| -|A ∩ B| =1. Then the coloring has a stable couple (p,q) where p and q from A, B outside intersection A ∩ B. We call the set of all outgoing edges of a vertex a bunch if all these edges are incoming edges of only one vertex. L.4 Let some vertex of AGW graph Γ have two incoming bunches. Then any coloring of Γ has a stable couple. stable couple (p,q) =1 The subset of states of maximal size such that every pair of its states is deadlock will be called an F-clique. A B Volkov, Whiller and Beal, Perrin simplification p q stable couple p q

5. Spanning subgraph (cluster) Let us call a subgraph S of the AGW graph Γ a spanning subgraph if to S belong all vertices of Γ and exactly one outgoing edge of any vertex. A maximal tree of the spanning subgraph S with root on cycle of S having no common edges with cycles from S is called a tree of S. The length of path from vertex p through edges of the spanning subgraph S to the root of its tree is called a level of p in S. Any spanning subgraph S consists of disjoint cycles and trees with roots on cycles; any tree and cycle of S is defined identically, the level of the cycle vertex is zero, the vertices of trees except root have positive level, the vertex of maximal positive level has no ingoing edge from S.

6. F-cliques and maximal level L. 5 Let N be a set of vertices of level n from some tree of the spanning set S of AGW graph Γ. Then via a coloring such that all edges of S have the same color, for any F-clique F holds |F ∩ N| ≤ 1. L. 6 Let spanning subgraph of an AGW graph Γ have no trees. Then Γ has a spanning subgraph with one vertex of maximal positive level. prqprq Only one from p r q could be in one F-clique

7. L. 6 The ancestors of a root on tree and cycle of a tree with vertices of maximal level form stable pair. F-cliques and maximal level p root … … cycle … a tree… b c p of maximal level, belongs to some F-clique F, a – letter on given edges pa L-1 =b, other states of F-clique F are mapped by a L-1 in cycles Then a m does not change states in cycles (image of F-clique F too except b ) -> ( b,c) stable a c … … cycle … b

8. L. 7 Let any vertex of an AGW graph Γ have no two incoming bunches. Then Γ has a spanning subgraph such that all its vertices of maximal positive level belong to one non-trivial tree. Find Maximal Tree p r … … cycle ….. … a b c tree

9. T. 3 Any AGW graph has a coloring with stable pairs. 1.The ancestors of a root on tree and cycle of a tree with all vertices of maximal level 2. The beginnings of two bunches having common end form stable pair. T. 4 Any AGW graph has a synchronizing coloring. Road coloring theorem for AGW graph Trahtman T.5 Let every vertex of strongly connected directed finite graph Γ have the same number of outgoing edges. Then Γ has synchronizing coloring if and only if the greatest common divisor of lengths of all its cycles is one.