One possible generalization of a Hamiltonian cycle in a triangulated plane graph is what could be called a Hamiltonian sphere: a collection of triangles within a simplicial complex in $\mathbb{R}^3$ that forms a surface homeomorphic to a sphere and which includes every vertex. For example, the triangulated surface of a cube is a Hamiltonian sphere for any one of the tetrahedralizations of the cube interior. And the notion could be generalized to arbitrary dimensions.

Has this concept been studied? I can imagine there are results specifying properties of the simplicial complex that guarantee it is Hamiltonian in the sense above. A trivial example is that the convex hull of points in general position constitute a Hamiltonian sphere for a triangulation of the hull interior into simplices.

One possible generalization of a Hamiltonian cycle in a triangulated plane graph is what could be called a Hamiltonian sphere: a collection of triangles within a simplicial complex in $\mathbb{R}^3$ that forms a surface homeomorphic to a sphere and which includes every vertex. For example, the triangulated surface of a cube is a Hamiltonian sphere for any one of the tetrahedralizations of the cube interior. And the notion could be generalized to arbitrary dimensions.

Has this concept been studied? I can imagine there are results specifying properties of the simplicial complex that guarantee it is Hamiltonian in the sense above. A trivial example is that the convex hull of points in convex position constitute a Hamiltonian sphere for a triangulation of the hull interior into simplices. (Here convex position means that all points are on the hull.)