Geodesic
Isomorphic components of direct products of bipartite graphs
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 231-248

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A standard result states the direct product of two connected bipartite graphs has exactly two components. Jha, Klavžar and Zmazek proved that if one of the factors admits an automorphism that interchanges partite sets, then the components are isomorphic. They conjectured the converse to be true. We prove the converse holds if the factors are square-free. Further, we present a matrix-theoretic conjecture that, if proved, would prove the general case of the converse; if refuted, it would produce a counterexample.
Keywords: direct product, tensor product, Kronecker product, bipartite graph 
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Hammack, Richard. Isomorphic components of direct products of bipartite graphs. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 2, pp. 231-248. http://geodesic.mathdoc.fr/item/DMGT_2006_26_2_a5/