I think i solved it (it's only for finite graphs), if anyone's intrested:
The d=1 case is easy (just two connected vertices).
We know that
, so there must be at least one cycle within. Since the graph is 2-colored, we know that there are no cycles of odd length, so the girth (size of the smallest cycle) is 4 (or greater).
If we put g=4 and
Now this tells us that graphs with d>=4 don't exist (We still need to find those with d=2,3).
The 2-regular is a square and 3-regular is a cube (so that there's no intersection between edges).