  Reed's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph
satisfies $\chi\leq \lceil\frac 12(\Delta+1+\omega)\rceil$; it is known to hold
for all claw-free graphs. In this paper we consider a local strengthening of
this conjecture. We prove the local strengthening for line graphs, then note
that previous results immediately tell us that the local strengthening holds
for all quasi-line graphs. Our proofs lead to polytime algorithms for
constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and
$O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster than the
best known algorithm for constructing a colouring that achieves the bound of
Reed's original conjecture.
