Hamiltonian triangular refinements and space-filling curves

We have introduced here the concept of Hamiltonian triangular refinement. For any Hamiltonian triangulation it is shown that there is a refinement which is also a Hamiltonian triangulation and the corresponding Hamiltonian path preserves the nesting condition of the corresponding space-filling curve. We have proved that the number of such Hamiltonian triangular refinements is bounded from below and from above. The relation between Hamiltonian triangular refinements and space-filling curves is also explored and explained

The vertex-adjacency dual of a triangulated irregular network has a Hamiltonian cycle

Triangulated irregular networks (TINs) are common representations of surfaces in computational graphics. We define the dual of a TIN in a special way, based on vertex-adjacency, and show that its Hamiltonian cycle always exists and can be found efficiently. This result has applications in transmission of large graphics datasets