, Volume 39, Issue 1-2, pp 145-154

The road-colouring problem

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Abstract

LetG be a finite directed graph which is irreducible and aperiodic. Assume each vertex ofG leads to at least two other vertices, and assumeG has a cycle of prime length which is a proper subset ofG. Then there exist two functionsr:GG andb:GG such that ifr(x)=y andb(x)=z thenxy andxz inG andyz and such that some composition ofr’s andb’s is a constant function.

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. I am grateful to Cornell University whose kind hospitality I enjoyed while working on this problem.