A Note on a Question of C. D. Savage

Abstract

Given a graph G and an orientation $\sigma$ of some of its edges, consider the graph $AO_\sigma(G)$ which is defined as follows: The vertices are the acyclic orientations of G which agree with $\sigma$, and two of these are adjacent if they differ only by the reversal of a single edge. $AO_\sigma(G)$ is easily seen to be bipartite. The purpose of this note is to show that it need not contain a Hamilton path even if both partite sets have the same cardinality. This answers a question of C. D. Savage [SIAM Rev., 39 (1997), pp. 605--629] and sheds new light onto two well-known open questions in the field of combinatorial Gray codes.