**Example 1.** Let $n=1$, $m=7$. Then $c=3$, and $G=C_3+C_4$ is a $2$-regular graph of order $7$ and chromatic number $3$, but is not isomorphic to $\mathbb Z_7$.

**Example 2.** Let $n=2$, $m=15$. Then $c=5$, as $G=3K_5$ is a $4$-regular graph of order $15$ and chromatic number $5$, but $\mathbb Z_{15}$ has chromatic number $3$.