A counterexample to a conjecture of Lovasz on the -coloring

Associated with every graph G of chromatic number is another graph G . The vertex set of G consists of all -colorings of G, and two -colorings are adjacent when they dier on exactly one vertex. According to a conjecture of Lovasz, this graph G must be disconnected. In this note we give a counterexample to this conjecture. In this paper we refute a conjecture of Lovasz. A complete account with the necessary background, and a proof of certain other instances of this conjecture can be found in [1]. For the purpose of the present note, no background is needed beyond elementary graph theory