So I have shown, that for all n $\in N $ there is a graph $G$ with n vertices such that the Greedy Algorithm will colour it in exactly 2 colours.
Further I have shown that for a Graph with diameter at least 3 there is an ordering of the vertices such that the Greedy Algorithm uses at least 3 colours. (both rather trivial tasks admittedly.)
Apparently these 2 rather basic insights should give me some more interesting property though, and I m wondering what that could be.