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

We construct and study a new class of compact hyperbolic 3-manifolds with totally geodesic boundary. This class extends the line of research developed in [1, 2, 3, 4] and exhibits a number of remarkable properties. The members of this class are described by triples of Eulerian cycles in 4-regular graphs. Two Eulerian cycles are said to be compatible if no pair of adjacent edges are consecutive in both cycles. If a 4-regular graph G contains a triple θ of pairwise compatible Eulerian cycles, we say that G is 3-Eulerian and θ is a framing of G. Each finite 3-vertex-connected simple 4-regular graph is 3-Eulerian [5]. Let G be a 3-Eulerian graph, and let θ be a framing of G. A polyhedral realization of the pair (G, θ) is a 2-dimensional polyhedron P (G, θ) obtained from G by attaching a 2-cell along each cycle in θ.