Coloring bipartite graphs with semi-small list size

TL;DR

This paper investigates asymmetric list coloring of bipartite graphs, establishing new bounds and invariants, especially for fixed list sizes, and proposes a unifying conjecture for bipartite graph choosability.

Contribution

It introduces an invariant for bipartite list coloring, improves bounds for list size 2, and unifies existing conjectures into a comprehensive framework.

Findings

Identified an invariant determining choosability in bipartite graphs.

Strengthened bounds for graphs with list size 2.

Proposed a unifying conjecture for bipartite graph coloring.

Abstract

Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex's list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting asymptotics, revealing an invariant quantity instrumental in determining choosability across most of the parameter space. By connecting this quantity to a simple question on independent sets of hypergraphs, we strengthen bounds when a part has list size 2. Finally, we state via our framework a conjecture on general bipartite graphs, unifying three conjectures of Alon-Cambie-Kang.