# $2n$-regular graphs with maximal chromatic number

Let $$n\geq 1$$ be an integer. Suppose $$m\geq 2n+1$$ is an integer. We construct the graph $$\mathbb{Z}_m = (\mathbb{Z}/m\mathbb{Z}, E_m)$$ where $$E_m=\big\{\{x,y\}:x, y \in \mathbb{Z}/m\mathbb{Z} \text{ and } \exists k\in\{1,\ldots n\}: (x+k = y \text{ or } y+k = x)\big\}.$$

It is easy to see that $$\mathbb{Z}_m$$ is a $$2n$$-regular graph on $$m$$ vertices.

Question. Suppose $$c$$ is the maximum chromatic number that a $$2n$$-regular graph on $$m$$ vertices can have, and let $$G=(V,E)$$ be a $$2n$$-regular graph with $$|V| = m$$ and $$\chi(G) = c$$. Does this imply that $$G\cong \mathbb{Z}_m$$?

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 $$3$$.