Automorphism Groups of Wreath Product Digraphs

We strengthen a classical result of Sabidussi giving a necessary and sufficient condition on two graphs, X and Y , for the automorphsim group of the wreath product of the graphs, Aut(X o Y ) to be the wreath product of the automorphism groups Aut(X) o Aut(Y ). We also generalize this to arrive at a similar condition on color digraphs. The main purpose of this paper is to revisit a well-known and important result of Sabidussi [17] giving a necessary and sufficient condition for the wreath product X oY (defined below) of two graphsX and Y to have automorphism group Aut(X)oAut(Y ), the wreath product of the automorphism group of X and the automorphism group of Y (defined below). We will both strengthen Sabidussi’s result and generalize it. First, Sabidussi only considered almost locally finite graphs X and finite graphs Y . (A graph is almost locally finite if the set of vertices of infinite degree is finite.) The condition that X be almost locally finite is needed for Sabidussi’s proof, but is clearly not needed in general. Indeed, note that X o Y , the complement of X o Y , has the same automorphism group as X o Y , X o Y = X̄ o Ȳ , but X̄ is not almost locally finite if X is infinite and almost locally finite. We will show that no restriction on X whatsoever is needed. We also weaken the requirement on Y : rather than requiring Y to be finite, we only require that Y not be isomorphic to a proper induced subgraph of itself. Next, since Sabidussi published his original paper, the wreath product of digraphs and color digraphs have also been considered in various contexts. We will give a necessary and sufficient condition for Aut(X oY ) = Aut(X)oAut(Y ) for a color digraph X and a color digraph Y , provided that X does not contain a specific forbidden digraph (which is infinite), and that Y is not isomorphic to a proper induced color subdigraph of itself. We then turn to the case where X is also finite and both X and Y are vertextransitive graphs (this is a common context in which Sabidussi’s result is applied), and show that if X and Y are not both complete or both edgeless, then there exist vertex-transitive graphs X ′ and Y ′ such that X o Y = X ′ o Y ′ and Aut(X o Y ) = Aut(X ′) o Aut(Y ′). Finally, the wreath product of Cayley graphs arises naturally in the study of the Cayley Isomorphism problem (definitions are provided in the third section, where this work appears). We show that if X and Y are CI-graphs of abelian groups G1 and G2, This research was supported in part by the National Science and Engineering Research Council of Canada.