Direct Product Factorization of Bipartite Graphs with Bipartition-reversing Involutions

Given a connected bipartite graph G, we describe a procedure which enumerates and computes all graphs H (if any) for which there is a direct product factorization G ∼= H × K2. We apply this technique to the problems of factoring even cycles and hypercubes over the direct product. In the case of hypercubes, our work expands some known results by Brešar, Imrich, Klavžar, Rall, and Zmazek [Finite and infinite hypercubes as direct products, Australas. J. Combin., 36 (2006), pp. 83–90, and Hypercubes as direct products, SIAM J. Discrete Math., 18 (2005), pp. 778–786].