The free energies of six-vertex models on a general domain D with various boundary conditions are investigated with the use of the n-equivalence relation, which help classify the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit (height, width) -> (infinity, infinity). It is derived that the free energies of the model on the domain D are classified through the densities of left/down arrows on the boundary. Specifically, the free energy is identical to that obtained by Lieb [Phys. Rev. Lett. 18, 1046 (1967); 19, 108 (1967); Phys. Rev. 162, 162 (1967)] and Sutherland [Phys. Rev. Lett 19, 103 (1967)] 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 calculation. The relation to the domino tiling (or dimer, or matching) problems is also noted. (C) 2008 American Institute of Physics.