$3$-manifolds represented by $4$-regular graphs with three Eulerian cycles

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