Geodesic
A vector space approach to the road coloring problem
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2010), pp. 14-20

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Let $G$ be a strongly connected, aperiodic, two-out digraph with adjacency matrix $A$. Suppose $A=R+B$ are coloring matrices: that is, matrices that represent the functions induced by an edge-coloring of $G$. We introduce a matrix $\Delta=\frac12(R-B)$ and investigate its properties. A number of useful conditions involving $\Delta$ which either are equivalent to or imply a solution to the road coloring problem are derived.
Keywords: digraph, synchronizing automaton, road coloring. 
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     author = {G. Budzban and Ph. Feinsilver},
     title = {A vector space approach to the road coloring problem},
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     pages = {14--20},
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     number = {1},
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}
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G. Budzban; Ph. Feinsilver. A vector space approach to the road coloring problem. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2010), pp. 14-20. http://geodesic.mathdoc.fr/item/IVM_2010_1_a2/