List Supermodular Coloring with Shorter Lists

In 1995, Galvin proved that a bipartite graph $G$ admits a list edge coloring if every edge is assigned a color list of length $\mathrm{\Delta }(G)$, 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.