List-Edge-Coloring with Bounded Maximum Degree

For a graph $G$, we show that if $mad(G)<m$, then ${\chi }_{\mathrm{\ell }}^{\mathrm{\prime }}(G)\le \mathrm{\Delta }+1$ where $m$ depends upon $\mathrm{\Delta }$ and ${\chi }_{\mathrm{\ell }}^{\mathrm{\prime }}(G)$ is the list-chromatic index of $G$. When $\mathrm{\Delta }\le 20$ the value of $m$ is close to $\frac{1}{2}\mathrm{\Delta }$, but as $\mathrm{\Delta }$ increases $m$ becomes asymptotic to about $\frac{1}{4}\mathrm{\Delta }+5$.