  We construct and study a new class $\mathscr{M}=\{\mathscr{M}_n\}_{n\ge 4}$
of compact hyperbolic $3$-manifolds with totally geodesic boundary. The members
of $\mathscr{M}_n$ are defined via triples of pairwise compatible Eulerian
cycles in $4$-regular $n$-vertex graphs. We show that each $M$ in
$\mathscr{M}_n$ is of Matveev complexity $n$ and has a unique minimal ideal
triangulation, which consists of $n$ tetrahedra. We exploit these properties to
show that $n!\,4^n > |\mathscr{M}_n| > n!$ for each sufficiently large
$n\in\mathbb{N}$.
