Hardness of finding a graph coloring given the optimal number of colors We know that graph coloring is NP-complete even in some special graph classes. On the other hand if someone tells you the exact value of the chromatic number of the input graph, is this problem remains NP-complete. In other words I am wondering if there are any result for the search version of the coloring problem (possibly on different graph classes).
Knowing the exact value of the chromatic number $\chi$ cannot help by more than a factor of $n$.  Since there are only $n$ possible values of $\chi$, you can 'guess' its value, i.e., run processes $P_1,\dots,P_n$, where $P_i$ runs an algorithm assuming $\chi=i$.  This whole scheme can find an optimum colouring in time at most $n$ times the time that it takes $P_\chi$ to find an optimum colouring.

On the other hand, if you're talking about parameterizing the running time by $\chi$, then it's a much more interesting question.  It's in FPT if you parameterize by $n-\chi$ S. Khot and V. Raman, ‘Parameterized Complexity of Finding Subgraphs with
Hereditary properties’.  If you parameterize by $\chi$ I would assume it's W[1]-hard.