List Supermodular Coloring with Shorter Lists

TL;DR

This paper generalizes key results in list edge coloring of bipartite graphs by extending theorems from Galvin, Borodin-Kostochka-Woodall, and Iwata-Yokoi to a broader supermodular coloring framework.

Contribution

It introduces a unified theorem that encompasses previous list coloring results for bipartite graphs and supermodular coloring, advancing the theoretical understanding.

Findings

Generalizes Galvin's list edge coloring theorem

Extends Borodin-Kostochka-Woodall's bounds to supermodular setting

Provides a unified framework for bipartite list coloring and supermodular coloring

Abstract

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.