Much work has gone into finding nontrivial upper bounds on the chromatic
number of a graph based on other graph invariants. Reed conjectured that
in fact $\chi(G) \leq \lceil (1/2)(\Delta+\omega+1) \rceil$ and proved
that the bound holds for the fractional chromatic number, even without the
round-up. I will discuss the proof of Reed's Conjecture, as it is widely
known, for the class of line graphs. I will also briefly mention some
recent work on Reed's Conjecture for the class of quasi-line graphs.
Joint work with Bruce Reed and A. Vetta