A quasicancellation property for the direct product of graphs

We are motivated by the following question concerning the direct product of graphs. If A×C ∼= B×C , what can be said about the relationship betweenA and B? If cancellation fails, what properties must A and B share?We define a structural equivalence relation∼ (called similarity) on graphs, weaker than isomorphism, for which A× C ∼= B× C implies A ∼ B. Thus cancellation holds, up to similarity. Moreover, if C is bipartite, then A× C ∼= B× C if and only if A ∼ B. We conjecture that the prime factorization of connected bipartite graphs is unique up to similarity of factors, and we offer some results supporting this conjecture. © 2009 Elsevier B.V. All rights reserved.