Cancellation of digraphs over the direct product

In 1971 Lovász proved the following cancellation law concerning the direct product of digraphs. If A, B and C are digraphs, and C admits no homomorphism into a disjoint union of directed cycles, then A × C ∼= B × C implies A ∼= B. On the other hand, if such a homomorphism exists, then there are pairs A ≁= B for which A×C ∼= B×C . This gives exact conditions on C that governwhether cancellation is guaranteed to hold or fail. Left unresolved was the question of what conditions on A (or B) force A × C ∼= B × C =⇒ A ∼= B, or, more generally, what relationships between A and C (or B and C) guarantee this. Even if C has a homomorphism into a collection of directed cycles, can there still be restrictions on A and C that guarantee cancellation? We characterize the exact conditions. We use a construction called the factorial A! of a digraph A. Given digraphs A and C , the digraph A! carries information that determines the complete set of solutions X to the digraph equation A × C ∼= X × C . We state the exact conditions under which there is only one solution X (namely X ∼= A) and that is the situation in which cancellation holds. © 2012 Elsevier Ltd. All rights reserved.