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:G →G andb:G →G such that ifr(x)=y andb(x)=z thenx →y andx →z inG andy ≠z and such that some composition ofr’s andb’s is a constant function.
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References
R. L. Adler, L. W. Goodwyn and B. Weiss,Equivalence of topological Markov shifts, Israel J. Math.27 (1977), 49–63.
J. W. Moon,Counting Labelled Trees, Canadian Mathematical Congress Monograph No.1, 1970.
G. L. O’Brien,Zero-inducing functions on finite abelian groups, to appear (1981).
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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.
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O’brien, G.L. The road-colouring problem. Israel J. Math. 39, 145–154 (1981). https://doi.org/10.1007/BF02762860
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DOI: https://doi.org/10.1007/BF02762860