An algorithm for road coloring

A. N. Trahtman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations


A coloring of edges of a finite directed graph turns the graph into a finite-state automaton. The synchronizing word of a 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 of uniform outdegree (constant outdegree of any vertex) is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph of uniform outdegree if the greatest common divisor of the lengths of all its cycles is one. The problem posed in 1970 has evoked noticeable interest among the specialists in the theory of graphs, automata, codes, symbolic dynamics as well as among the wide mathematical community. A polynomial time algorithm of O(n3) complexity in the worst case and quadratic in the majority of studied cases for the road coloring of the considered graph is presented below. The work is based on the recent positive solution of the road coloring problem. The algorithm was implemented in the freeware package TESTAS.

Original languageEnglish
Title of host publicationCombinatorial Algorithms - 22nd International Workshop, IWOCA 2011, Revised Selected Papers
Number of pages12
StatePublished - 2011
Event22nd International Workshop on Combinatorial Algorithms, IWOCA 2011 - Vancouver, BC, Canada
Duration: 20 Jul 201122 Jul 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7056 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference22nd International Workshop on Combinatorial Algorithms, IWOCA 2011
CityVancouver, BC

Bibliographical note

cited By 6


  • algorithm
  • deterministic finite automaton
  • graph
  • road coloring
  • synchronization


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