  We prove that $\min\{\chi(G), \chi(H)\} - \chi(G\times H)$ can be arbitrarily
large, and that if Stahl's conjecture on the multichromatic number of Kneser
graphs holds, then $\min\{\chi(G), \chi(H)\}/\chi(G\times H) \leq 1/2 +
\epsilon$ for large values of $\min\{\chi(G), \chi(H)\}$.
