Is the minimal number of colors needed to color a graph some fixed number?
The problem is always in $\Delta_2 = \Sigma_2 \cap \Pi_2$, as you mention. For some values of 2015 it is easier: deciding whether a graph has chromatic number 3 is NP-complete, and deciding whether a graph has chromatic number 2 or 1 is in P (since you can check whether a graph is bipartite in polytime). 

Since deciding whether a graph has chromatic number 3 is NP-hard, so is deciding whether it has chromatic number 2015, by reduction: add a clique of size 2012, and connect it to all vertices; this increases the chromatic number by exactly 2012. So there is no hope for a polynomial time algorithm.