  Given a graph $G$ and a family $\mathcal{G} = \{G_1,\ldots,G_n\}$ of
subgraphs of $G$, a transversal of $\mathcal{G}$ is a pair $(T,\phi)$ such that
$T \subseteq E(G)$ and $\phi: T \rightarrow [n]$ is a bijection satisfying $e
\in G_{\phi(e)}$ for each $e \in T$. We call a transversal Hamiltonian if $T$
corresponds to the edge set of a Hamiltonian cycle in $G$. We show that, under
certain conditions on the maximum degree of $G$ and the minimum degrees of the
$G_i \in \mathcal{G}$, for every $\mathcal{G}$ which contains a Hamiltonian
transversal, the number of Hamiltonian transversals contained in $\mathcal{G}$
is bounded below by a function of $G$'s maximum degree. This generalizes a
theorem of Thomassen stating that, for $m \geq 300$, no $m$-regular graph is
uniquely Hamiltonian. We also extend Joos and Kim's recent result that, if $G =
K_n$ and each $G_i \in \mathcal{G}$ has minimum degree at least $\frac{n}{2}$,
then $\mathcal{G}$ has a Hamiltonian transversal: we show that, in this
setting, $\mathcal{G}$ has exponentially many Hamiltonian transversals.
Finally, we prove analogues of both of these theorems for transversals which
form perfect matchings of $G$.
