A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

In 1960, Sabidussi conjectured that if a graph G is isomorphic to the lexicographic product G[G], then the wreath product of Aut(G) by itself is a proper subgroup of Aut(G[G]). A positive answer is provided by constructing an automorphism Psi of G[G] which satisfies: for every vertex x of G, there is an infinite subset I (x) of V (G) such that Psi({x} x V(G)) = I (x) x V (G). (C) 2007 Elsevier B.V. All rights reserved.