  The free energies of six-vertex models on general domain D with various
boundary conditions are investigated with the use of the n-equivalence relation
which classifies the thermodynamic limit properties. It is derived that the
free energy of the six-vertex model on the rectangle is unique in the limit in
which both the height and the width goes to infinity. It is derived that the
free energies of the model on D are classified through the densities of
left/down arrows on the boundary. Specifically the free energy is identical to
that obtained by Lieb and Sutherland with the cyclic boundary condition when
the densities are both equal to 1/2. This fact explains several results already
obtained through the transfer matrix calculations. The relation to the domino
tiling (or dimer, or matching) problems is also noted.
