CITATION of the publications


ROAD COLORING, ArXiv downloads



Listings & Recognitions of results 

From Encyclopedia Britannica

         Avraham Trahtman Israeli mathematician

born Feb. 10, 1944, Kalinovo, Sverdlovsk oblast, U.S.S.R. [now in Russia]


The astonishing news emerged in 2008 that in September 2007 a 63-year-old Israeli mathematician, Avraham Trahtman, had solved a long-standing problem in graph theory. The road-colouring conjecture, as it was known before being solved by Trahtman, was first suggested in 1970 by the Israeli-American mathematician Benjamin Weiss and the American mathematicians Roy L. Adler and L. Wayne Goodwyn. The theorem concerns a special type of graph, or network, that fulfills certain conditions. The network must have a finite number of vertices (specific locations, or points) and directed edges (one-way paths), be strongly connected (a path must exist from any vertex a to any other vertex b and a path from b to a), and aperiodic (essentially, the cycles, or complete routes following different directions, must be independent). The road-colouring theorem asserts that for such a network, there always exists a synchronized colouring, or method of labeling the edges, to create a map with a simple set of directions, possibly involving many repetitions of the directions, that will lead from any starting point to any other given point. In other words, by following simple directions, such as to take a “red-blue-red” path, it is possible to start from any location and be certain to end up at the desired destination. The existence of such “universal maps” has theoretical implications for real-life problems in computer science, though having a universal map is not the same as having the most efficient route for any individual case.

Trahtman earned an undergraduate degree (1967) and a graduate degree (1973) in mathematics from Ural State University, in Sverdlovsk (nowYekaterinburg, Russia). He taught in that same city at the Ural State Technical University (1969–84) and at the Sverdlovsk Pedagogical University (1991–92) before immigrating to Israel in 1992. Like many of the recent immigrants following the breakup of the Soviet Union, Trahtman had difficulty finding an academic position, and he first accepted work as a security guard and lectured (1994–95) part-time in the pre-education department at Hebrew University in Jerusalem. In 1995 Trahtman obtained a professorship at Bar-Ilan University in Ramat Gan, near Tel Aviv.

Trahtman’s solution is notable for its brevity: at less than eight pages it is extremely concise and considered quite elegant. The proof is also rather noteworthy in that most major mathematical breakthroughs come from individuals before the age of 40, or even 30. Trahtman proved, however, that age was not a barrier to solving a nearly 40-year-old mathematical conundrum.


Encyclopedia Britannica.


Encyclopedia Britannica Australia.


Wikipedia (English, German, Russian, France …)


MacTutor History of Mathematics, Biographies

Wolfram Research Mathematica


Lambert M. Surhone, Miriam T. Timpledon, Susan F. Marseken

 Road Coloring Problem: Theorem, Conjecture, Synchronization,

Instruction, Symbolic Dynamics, Maze, Avraham Trahtman

Publisher: Vdm Verlag Dr. Mueller Ag & Co. Kg (2010)


Israeli Mathematicians: Adi Shamir, Giulio Racah, Saharon Shelah,

Zlil Sela, Robert Aumann, Michael O. Rabin, Oded Schramm, Avraham Trahtman

Publisher: Books LLC (Editor) (September 15, 2010) ISBN-10: 1156843480


Den sidste gode mand A.J. kazinski, Politiken. 2010 (18 translations)

The Last Good Man  A.J. Kazinski Translated by: Tiina Nunnally Simon @Schuster 2010

Die Auserwählten von A.J. kazinski, Krimi-couch, 2011

А. Й. Казински. Последний праведник. : Астрель, 2012.


Arthur Benjamin, Gary Chartrand & Ping Zhang

The Fascinating World of Graph Theory

Popular Math.,, 360, 2015


New Scientist, 186 (2495), 2005, London
Pearce F., Pyramids of dew 52-53

Наука и жизнь 2020 № 2

Универсальный алгоритм



Published: April 24, 1988 DAVID K. SHIPLER

Excerpts from some recently published letters

Ogonyok, No. 15 April 1988  A. Trakhtman


Timeline Israel 1961-1995 (1970)
Timelines (1970)

Who'sWho in the World. 2006-2008
Marquis Who's Who, 23th-28th Ed., NY

Dictionary of International Biography
33th Ed., London, 2006

The Times (4 March 2009). M du Sautoy, Sexy maths: ditch the GPS, just follow the colour code,

The Guardian (Friday 21 March 2008). A Heller, Security guard solves 38-year-old maths poser,

The Telegraph (21 March 2008). R Highfield, Directions from anywhere: maths problem solved,

The. Jerusalem Post (8 February, 2008) J Siegel-Itzkovich, Russian immigrant solves math puzzle,

Haaretz Newspaper (20 March 2008).Israeli ex-security guard solves 38-year-old math problem
Math in Media, AMS, 03.08
Associated Press, Aron Heller, 20.03.08

The Canadian Press, Aron Heller, 20.03.08

Ediot ahronot, N. Mozgovaja, 06.02.08
International Herald Tribune, 20.03 08

BBC News, spanish, Mundo, 12.02.08

The Herald, 22.03.08

Math Gateway of AMS, 25.02.08

 IsraelNN, AruzSheva, Hillel Fendel, 07.02.08

The Epoch Times, in China, 23.03.08
Physics Today, 21.03.08
 Millennium Mathematics Project, University of Cambridge.

Hacofe, Hadar Rabid, 24.02.08
Evreiskij mir USA, 08.02.08
BecomingJewish, 10.02.08
 IsRealli, Israel official, 14.02.08
Israel 21C, Karin Kloosterman. 18 02 2008.

Neue Zürcher Zeitung, George Szpiro, Wie man auf Irrwegen ans Ziel gelangt, 30.03.2008

Kristeling Dagblat, 12.02.08
Red Cubana, Cuba, 13.02.08
 CloudWatcher, 09.02.08

ProphecyNewsWatch, 09.02.08
Forex Times, 18.02.08, 09.02.08
 Itogi, N9, Astakhova, 06.03.08
Nauka v sviti, Ukraina 22.03.08
 Izvestia, D. Varlamova, 04.04.08
Novosti, 10.04.08
 REN TV, 04.06.08
Belfast Telegraph 21.03.08
Contemporary Centrist, 21.03.08
 Deccan Herald 21.03.08
Izmirdeyasam, 22.03.08

and other information in dozen of languages





Prof.  Dr.  Avraham Trahtman

Academic interests

Algebra, finite automata, water problems, algorithms, computing.


1962-1967 Ural State University, Mathematics, degree II, 1967.
1970-1972 Ural State University, Mathematics, degree III,1973.

Work experience

1995- 2012 Bar-Ilan University, Department of Mathematics and Computer Science.
1994-1995 Hebrew University - Lecturer, Pre-education Department

1991-1992 Assistant professor, Sverdlovsk Pedagogical University, Department of Mathematics.
1969-1984 From assistant to associate professor, Department of Computational Mathematics, Ural Technical University.



Creation of package TESTAS for verification of local testability and its generalizations (threshold, right, left, bilateral, piecewise and other) for both transition graphs of automata and transition semigroups, for checking synchronizability and finding synchronizing words, for finding transition semigroup of automaton.
The C/C++ compact package TESTAS is mentioned on home pages of several prominent CONFERENCES among the embedded systems for manipulating automata.


English, German, Hebrew, Russian, Polish.