Unit-distance graphs in Minkowski metric spaces

The Unit-Distance Graph problem in Euclidean plane asks for the minimum number of colors, so that each point on the Euclidean plane can be assigned a single color with the condition that the points at unit distance apart are assigned different colors. It is well known that this number is between 4 and 7, but the exact value is not known. Here this problem is generalized to Minkowski metric spaces and once again the answer is shown to be between 4 and 7. In extreme special cases where the unit circle is a parallelogram or a hexagon the answer is shown to be exactly 4.