  The exact finite-size corrections to the free energy $F$ of the dimer model
on lattice $\mathcal{M} \times \mathcal{N}$ with cylindrical boundary
conditions have been derived for three cases where the lattice is completely
covered by dimers: $\mathcal{M} = 2M$, $\mathcal{N} = 2N$; $\mathcal{M} = 2M -
1$, $\mathcal{N} = 2N$; and $\mathcal{M} = 2M$, $\mathcal{N} = 2N - 1$. For
these types of cylinders, ratios $r_p(\rho)$ of the $p$th coefficient of $F$
have been calculated for the infinitely long cylinder (${\mathcal M}
\rightarrow \infty$) and infinitely long strip (${\mathcal N} \rightarrow
\infty$) at varying aspect ratios. As in previous studies of the dimer model on
the rectangular lattice with free boundary conditions and for the Ising model
with Brascamp-Kunz boundary conditions, the limiting values $p \to \infty$
exhibit abrupt anomalous behaviour of ratios $r_p(\rho)$ at certain values of
$\rho$. These critical values of $\rho$ and the limiting values of the
finite-size expansion coefficient ratios vary between the different models.
