are there any results on the number of graphs on n vertices that has chromatic number=k ?


I mean how many graphs on n vertices has x(G)=2,x(G)=3,......,x(G)=n-1 ?


**update**:
infact equivalently, is there a way to count unique k-partite graphs?
after that to get x(G)=k, just subtract #k-partite-#(k-1)-partite.
I've been almost able to count number with x(G)=2 using a much easier way but the problem remains for k>=3


This is an expression I got for k=2 but needs to be verified:
$|\chi(G)=2|=n^{n-2}\frac{n+1}{2}=\frac{n^{n-1}+n^{n-2}}{2}$



Counting graphs on n vertices by chromatic number
The formula for chromatic number 2 can't quite be right, as it gives $3/2$ when $n=2$. [Number](http://oeis.org/A000683) of 2-colored labeled graphs on $n$ nodes is tabulated at the Online Encyclopedia of Integer Sequences. The listing there agrees with the formula here for $n=3$ and $n=4$ but gives 360, not 375, for $n=5$. The page also references 


F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, table 1.5.1, column 2 (divided by 2) and 


R. C. Read, The number of k-colored graphs on labeled nodes, Canad. J. Math., 12 (1960), 410-414.


