# The Prism of the Acyclic Orientation Graph is Hamiltonian

### Abstract

Every connected simple graph $G$ has an acyclic orientation. Define a graph ${AO}(G)$ whose vertices are the acyclic orientations of $G$ and whose edges join orientations that differ by reversing the direction of a single edge. It was known previously that ${AO}(G)$ is connected but not necessarily Hamiltonian. However, Squire proved that the square ${AO}(G)^2$ is Hamiltonian. We prove the slightly stronger result that the prism ${AO}(G) \times e$ is Hamiltonian.

If $G$ is a mixed graph (some edges directed, but not necessarily all), then ${AO}(G)$ can be defined as before. The graph ${AO}(G)$ is again connected but we give examples showing that the prism is not necessarily Hamiltonian.