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

Abstract

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 ${\chi }_{\ell }(G)\le \Delta (G)-1$.

24 pages, 5 figures