On uniqueness of prime bipartite factors of graphs

It has long been known that the class of connected nonbipartite graphs (with loops allowed) obeys unique prime factorization over the direct product of graphs. Moreover, it is known that prime factorization is not necessarily unique in the class of connected bipartite graphs. But any prime factorization of a connected bipartite graph has exactly one bipartite factor. Moreover, empirical evidence suggests that any two prime factorings of a given connected bipartite graph have isomorphic bipartite factors. This prompts us to conjecture that among all the different prime factorings of a given connected bipartite graph, the bipartite factor is always the same. The present paper proves that the conjecture is true for graphs that have a K2 factor. (Even in this simple case, the result is surprisingly nontrivial.) Further, we indicate how to compute all possible prime factorings of such a graph. In addition, we show how the truth of the conjecture (in general) would lead to a method of finding all distinct prime factorings of any connected bipartite graph. To accomplish this, we prove the following preliminary result, which is the main technical result of the paper: Suppose A × B is connected and bipartite, and B is the bipartite factor. If A × B admits an involution that reverses partite sets, then B also admits such an involution.