Maximum Chromatic number of Cayley Graphs with large degree
Erdős and Gallai showed in their paper Solution of a problem of Dirac then the chromatic number of a non-complete regular graph is at most $\frac{3}{5}n$.
Caccetta and Pullman constructed in their paper Regular graphs with prescribed chromatic number connected $k$-chromatic regular graphs on $n$ vertices for all $k &gt; 1$ and $n \geq \frac{5}{3}k$. You can check their construction to see whether it can be adapted to give Cayley graphs.