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.