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In the paper Graphical complexity of products of permutation groups, M.Grech, A. Jez, A. Kisielewicz have proved that the direct product ofautomorphism groups of edge-colored graphs is itself the automorphism groups ofan edge-colored graph. In this paper, we study the direct product of twopermutation groups such that at least one of them fails to be the automorphismgroup of an edge-colored graph. We find necessary and sufficient conditions forthe direct product to be the automorphism group of an edge-colored graph. Thesame problem is solved for the edge-colored digraphs.

Decompositions of the authomorphism groups of edge-colored graphs into the direct product of permutation groups