To which complexity class does this language belong? I was thinking about which class this language belongs:
$L =\{ \langle G,k \rangle \mid G $ is a graph, $k$ is a natural number and $k$ is the chromatic number of $G\}$

I thought of $L$ as (1) " there is no coloring of k-1 colors" and (2) "there is coloring of $k$ colors". Now, (1) is coNP and (2) is NP-complete so I assume that this language is neither in NP nor in coNP, but I didn't find to which class it belongs.
Any help will be welcomed.

Thanks
(as pointed out by Robin the problem is in DP...)

...and it is also DP-complete.

In fact, Jörg Rothe has shown that this even holds for fixed k=4: Jörg Rothe: Exact complexity of Exact-Four-Colorability. Inf. Process. Lett. (IPL) 87(1):7-12 (2003)