Multiplicativity of Oriented Cycles

We complete the characterization of those oriented cycles C which have the property that the class of digraphs not homomorphic to C is closed under taking the (categorical) product. This is related to Hedetniemi′s conjecture on the chromatic number of the product of undirected graphs. Our main tool is a result concerning the existence of homomorphisms between oriented paths which preserve the initial and terminal vertices. This tool is used to prove that for those cycles C we are interested in, any two oriented paths not homomorphic to C have a common preimage also not homomorphic to C. This "common preimage theorem" implies the multiplicativity of our cycles.