It is given that we have N points arranged along a circle and $N \geq k(k+1)$. $G_k$ is the graph we get after connecting every point out of $N$ on the circle to it's $k$ nearest neighboring points in all directions of the circle.
I have to prove that $$ \chi(G_k) = (k+1) \text{ if n is divisible by k+1}$$ $$ =\text{ }(k+2) \text{ else } $$
Here, $\chi$ denotes the chromatic number of a graph, although I am aware that it essentially denotes the different colors for graph coloring , I am not sure as to how I would get the desired result.
Any help is greatly appreciated.