Abstract
The road coloring problem has been open for some 25 years. This paper shows how algebraic methods, specifically semigroup theory, can be used to both generalize and shed light on the problem. Given a strongly connected digraph, the notion of a coloring semigroup is defined. The main result shows that the existence of a coloring semigroup whose kernel is a minimum rank right group of rank t implies the digraph is periodic of order t.
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Budzban, G. Semigroups and the Generalized Road Coloring Problem. Semigroup Forum 69, 201–208 (2004). https://doi.org/10.1007/s00233-004-0129-x
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DOI: https://doi.org/10.1007/s00233-004-0129-x