A graph is weakly perfect if the clique number equals the chromatic number, i.e. $\omega(G)=\chi(G)$. Deciding membership is NP-complete according to the paper.
Because of the inequality $\omega(G) \le \chi(G)$ a $k$-clique is certificate that there isn't smaller than $k$ coloring.
What is the complexity of deciding if a weakly perfect graph is $k$-colorable?
Appears to me it might not be NP-complete because the certificate will involve co-NP.
Also it might not be in P since it will decide coloring graphs for which the inequality happens "by chance" and possibly by adding disjoint cliques might be improved.
Searching the web found algebraic constructions of weakly perfect. In all papers coloring was polynomial via closed form.
Related to this.