List-coloring claw-free graphs with $\Delta$-1 colors

We prove that if $G$ is a quasi-line graph with $\Delta(G)>\omega(G)$ and $\Delta(G)\ge 69$, then $\chi_{OL}(G)\le \Delta(G)-1$. Together with our previous work, this implies that if $G$ is a claw-free graph with $\Delta(G)>\omega(G)$ and $\Delta(G)\ge 69$, then $\chi_{\ell}(G)\le \Delta(G)-1$.
Comment: 24 pages, 5 figures