What is equation of straight line for a ideal blackbody on the color color diagram? The color-color diagram is a plot between $U-B$ vs $B-V$ values. 

For a ideal blackbody, what is the equation of this line?

In other words, what does the linear relation between $U-B$ and $B-V$ look like?
Use Planck's law: $$F(\lambda,T) = c  \lambda^{-5}  (e^{d/(\lambda  T)} - 1)^{-1}$$.

Since magnitude is defined as, $M = -2.5 \log_{10}(F) + \textrm{constant}$, the color index $A-B$ is
$$(A - B) = -2.5 \log_{10}(F(\lambda_A,T)/F(\lambda_B,T))$$.

To answer your question, use an appropriate effective wavelength of $U,B,V$. Link $U-B$ and $B-V$ via temperature, you should get $U-B = f(B-V)$.