  Fay, Hurlbert and Tennant recently introduced a one-player game on a finite
connected graph $G$, which they called cup stacking. Stacks of cups are placed
at the vertices of $G$, and are transferred between vertices via stacking
moves, subject to certain constraints, with the goal of stacking all cups at a
single target vertex. If this is possible for every target vertex of $G$, then
$G$ is called stackable. In this paper, we prove that if $G$ admits a Hamilton
path, then $G$ is stackable, which confirms several of the conjectures raised
by Fay, Hurlbert and Tennant. Furthermore, we prove stackability for certain
powers of bipartite graphs, and we construct graphs of arbitrarily large
minimum degree and connectivity that do not allow stacking onto any of their
vertices.
