SIAM J. DISCRETE MATH. c  2010 Society for Industrial and Applied Mathematics Vol. 23, No. 4, pp. 2042–2052 DIRECT PRODUCT FACTORIZATION OF BIPARTITE GRAPHS WITH BIPARTITION-REVERSING INVOLUTIONS ∗ GHIDEWON ABAY-ASMEROM † , RICHARD H HAMMACK † , CRAIG E. LARSON † , AND DEWEY T. TAYLOR † Abstract. 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 × K 2 . 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 Breˇ sar, Imrich, Klavˇ zar, 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]. Key words. graph direct product, graph factorization, bipartite graphs, hypercubes AMS subject classification. 05C60 DOI. 10.1137/090751761 1. Introduction. A graph G =(V (G),E(G)) in this paper is finite and may have loops, but not multiple edges. The direct product of two graphs G and H is the graph G × H whose vertex set is the Cartesian product V (G) × V (H ) and whose edges are E(G × H )= {(g,h)(g ′ ,h ′ ): gg ′ ∈ E(G), hh ′ ∈ E(H )}. The graphs G and H are called factors of the product. If I is the graph with one vertex and one loop, then I × G ∼ = G for any graph G. A graph G is said to be prime with respect to × if whenever G ∼ = H × K, one factor is isomorphic to I and the other is isomorphic to G. A fundamental result due to McKenzie [7] (see also Imrich [6]) implies that any connected non-bipartite graph has a unique prime factorization over the direct product. It is known that bipartite graphs are not uniquely prime factorable, but the ways that they can decompose into prime factors is largely unexplored. This paper addresses the ways that a bipartite graph G can be factored as a product G ∼ = H × K 2 of a graph H with the complete graph K 2 . This is in some ways analogous to factoring an even integer g into a product g = h · 2, except that for graphs the factorization need not be unique. For example, Figure 1 shows that the 10-cycle G = C 10 can be prime factored as G ∼ = H × K 2 where H can be either the path P 5 with loops at each end, or the cycle C 5 . As we shall see, in general such graphs H arise in a simple way from the automorphism conjugacy classes of the involutions that reverse the bipartition of G. We will apply our results to the problem of extracting K 2 factors from even cycles and hypercubes. We note that our current paper falls partly under the umbrella of [1]. Given arbitrary graphs H and K, with K bipartite, [1] classifies all the graphs H ′ for which H × K ∼ = H ′ × K. Replacing K with K 2 would seem to cover the topic of the current paper. However [1] employs a somewhat complex construction called the factorial of H , and thus in general the graphs H ′ appear to be difficult to compute. By contrast ∗ Received by the editors March 6, 2009; accepted for publication (in revised form) September 15, 2009; published electronically January 15, 2010. http://www.siam.org/journals/sidma/23-4/75176.html † Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Box 842014, Richmond, VA 23284-2014 (ghidewon@vcu.edu, rhammack@vcu.edu, clarson@vcu.edu, dttaylor2@vcu.edu). 2042