The road coloring problem

TL;DR

This paper presents a positive solution to the longstanding road coloring problem, demonstrating how to find a synchronizing coloring for certain directed graphs.

Contribution

The paper provides the first complete proof that the road coloring problem has a positive solution for all applicable graphs.

Findings

The road coloring problem is solvable for all strongly connected graphs with constant outdegree and gcd of cycle lengths equal to one.

The solution advances understanding of synchronizing automata and graph colorings.

The problem, previously open for over 30 years, is now fully resolved.

Abstract

The synchronizing word of 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 deterministic finite automaton possessing a synchronizing word. The road coloring problem is a problem of synchronizing coloring of 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 a noticeable interest among the specialists in theory of graphs, deterministic automata and symbolic dynamics. The problem is described even in "Wikipedia" - the popular Internet Encyclopedia. The positive solution of the road coloring problem is presented.