For an undirected graph $G$, let us denote $\gamma(G)$ the chromatic number of $G$. Let us define for positive integer $n$, the language $$ L_n = \{\langle G\rangle : \gamma(G) = n\}. $$ This is the language of undirected graphs with chromatic number $n$. I think $L_n \in \mathrm{P}^{\mathrm{NP}}$, for any $n$.
My argument is the following. We can look at the language, $$ \mathrm{CHROMNUM} = \{\langle G, k\rangle : \gamma(G) \leq k\}. $$ It is clear that $\mathrm{CHROMNUM} \in \mathrm{NP}$, since we can verify an instance just by displaying an at most $k$-coloring of $G$.
Thus, consider oracle Turing machine $M^{\mathrm{CHROMNUM}}_n$, which on input $G$, simply checks that $\langle G, n \rangle \in \mathrm{CHROMNUM}$ and that $\langle G, k \rangle \not \in \mathrm{CHROMNUM}$ for all $k \leq n$.
I've just read about $\mathrm{P}^{\mathrm{NP}}$, so I wanted to be sure that this kind of argument is correct.