Abstract
In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jeż, and A. Kisielewicz have proved that the direct product of automorphism groups of edge-colored graphs is itself the automorphism groups of an edge-colored graph. In this paper, we study the direct product of two permutation groups such that at least one of them fails to be the automorphism group of an edge-colored graph. We find necessary and sufficient conditions for the direct product to be the automorphism group of an edge-colored graph. The same problem is settled for the edge-colored digraphs.
Highlights
For permutation groups (A, V ), (B, W ), the direct product of A and B is a permutation group (A × B, V × W ) with the action given by (a, b)(x, y) = (a(x), b(y)).The study of the direct product of automorphism groups of graphs was initiated by G
By DG(k) we denote the class of the automorphism groups of k-edge-colored digraphs, and by DGR the union of all the classes DG(k)
The main results is Theorem 3.2 characterizing the conditions under which the direct product of two arbitrary permutation groups belongs to GR
Summary
Imrich [13], have described the conditions under which the direct product of regular permutation groups that are automorphism groups of graphs is itself the automorphism group of a graph By DG(k) we denote the class of the automorphism groups of k-edge-colored digraphs, and by DGR the union of all the classes DG(k) (which in Wielandt’s terminology is the class of 2-closed groups). The main general problem is to determine which permutation groups are the automorphism groups of edge-colored graphs. The main results is Theorem 3.2 characterizing the conditions under which the direct product of two arbitrary permutation groups belongs to GR
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