Let $Q_n$ be the Queen's graph with $n^2$ vertices. I was asked to show that $\chi(Q_n) = n$ whenever $6$ and $n$ are coprime. I have failed to provide a coloring that proves this for the general case.
It is straightforward to see that any diagonal, column or row in the "board" is a $K_n$ graph, and therefore $\chi(Q_n) \geq n$. But I cannot see how the fact that $n$ is coprime with $6$ restricts the optimal coloring to $n$.
I'd appreciate any hint or answer. Thanks in advance.