Geodesic
A cancellation property for the direct product of graphs
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 179-184.

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Given graphs A, B and C for which A×C ≅ B×C, it is not generally true that A ≅ B. However, it is known that A×C ≅ B×C implies A ≅ B provided that C is non-bipartite, or that there are homomorphisms from A and B to C. This note proves an additional cancellation property. We show that if B and C are bipartite, then A×C ≅ B×C implies A ≅ B if and only if no component of B admits an involution that interchanges its partite sets.
Keywords: graph products, graph direct product, cancellation 
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Hammack, Richard. A cancellation property for the direct product of graphs. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 179-184. http://geodesic.mathdoc.fr/item/DMGT_2008_28_1_a12/

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