Cover numbers by graph families bounded by certain graph parameters

The cover number of a graph by a graph class $\mathcal P$ is the least number of $\mathcal P$-graphs necessary to cover its edges. A classical theorem of Harary, Hsu and Miller gives an exact formula for the cover number by the class of graphs with chromatic number at most $k$. We investigate analogous questions for the case of the fractional chromatic number $\chi_f$ and the local chromatic number $\psi$.
We prove that an analogous formula cannot hold in the case of the cover number by graphs of fractional chromatic number at most $\beta$, and find a lower and an upper bound, that gives rise to interesting asymptotic questions. We also investigate this cover number for small specific graphs. In the case of the cover number by graphs with local chromatic number at most $k$, we find an upper bound in terms of $\psi$, and a lower bound in terms of $\omega$.