Abstract

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.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.