Consider the graph coloring problem: given an undirected graph $G$ and a natural number $n$ return yes if we can color the graph with n different colors and no otherwise.
I am able to design a deterministic Turing machine that would solve it with a greedy approach trying the different combinations of coloring with the $n$ colors. I think this takes exponential time although I'm not sure how to proof it.
However, I am not able to conceive a nondeterministic Turing machine that would do so. Can someone guide me designing an algorithm for this machine?