Direct products of automorphism groups of graphs

Abstract

Abstract:In this article we study the product action of the direct product of automorphism groups of graphs. We generalize the results of Watkins [J. Combin Theory 11 (1971), 95--104], Nowitz and Watkins [Monatsh. Math. 76 (1972), 168--171] and W. Imrich [Israel J. Math. 11 (1972), 258--264], and we show that except for an infinite family of groups S n ×S n , n ≥ 2 and three other groups D 4 ×S 2 , D 4 ×D 4 and S 4 ×S 2 ×S 2 , the direct product of automorphism groups of two graphs is itself the automorphism group of a graph.

Direct product of automorphism groups of digraphs

“…Our main result is Theorem 2.1 that says that, given two permutation groups (A, V ) and (B, W ) that have digraph representations, their direct product (A × B, V × W ) also has a digraph representation, unless A × B is one of the four exceptional groups D 4 × S 2 , D 4 ×D 4 , S 4 ×S 2 ×S 2 , C 3 ×C 3 , or a member of the infinite family of groups S n ×S n , n ≥ 2. It is a digraph counterpart of Theorem 2.10 of [8] by Grech for undirected graphs.…”

“…We begin by extending Theorem 2.10 of [8] by Grech for undirected graphs to directed graphs. Lemma 2.2.…”

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

“…This result was a contribution to the description of all regular automorphism groups of graphs, which has been completed in 1978 by C. Godsil [5] for graphs, and in 1980 by L. Babai [2] for digraphs. The above results in [19,13] have been extended to arbitrary permutation groups in [6], where the description of the conditions, under which the direct product of automorphism groups of graphs is itself an automorphism group of a graph, is given. In [8], the same is done for digraphs.…”

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

“…The results given in [19,13] are extended to arbitrary permutation groups in [6], where the description of the conditions, under which the direct product of automorphism groups of graphs is itself an automorphism group of a graph, is given. In [8], the same is done for digraphs.…”