  In 1995, Galvin proved that a bipartite graph $G$ admits a list edge coloring
if every edge is assigned a color list of length $\Delta(G)$, the maximum
degree of the graph. This result was improved by Borodin, Kostochka and
Woodall, who proved that $G$ still admits a list edge coloring if every edge
$e=st$ is assigned a list of $\max\{d_{G}(s), d_{G}(t)\}$ colors. Recently,
Iwata and Yokoi provided the list supermodular coloring theorem, that extends
Galvin's result to the setting of Schrijver's supermodular coloring. This paper
provides a common generalization of these two extensions of Galvin's result.
