Extracting list colorings from large independent sets
Hal Kierstead, Landon Rabern

TL;DR
This paper applies the Kernel Lemma to derive new results in online list coloring, including an Ore-degree version of Brooks' Theorem, bounds for triangle-free graphs, and a characterization of Gallai trees.
Contribution
It introduces a novel application of the Kernel Lemma to online list coloring, establishing new bounds and characterizations in graph theory.
Findings
Graphs with Ore-degree ≥18 and clique number ≤ half of Ore-degree are online list-colorable.
Upper bound for online list-coloring of triangle-free graphs: Δ+1 minus a logarithmic term.
Gallai trees characterized as connected graphs with no large independent set incident to many edges.
Abstract
We take an application of the Kernel Lemma by Kostochka and Yancey to its logical conclusion. The consequence is a sort of magical way to draw conclusions about list coloring (and online list coloring) just from the existence of an independent set incident to many edges. We use this to prove an Ore-degree version of Brooks' Theorem for online list-coloring. The Ore-degree of an edge in a graph is . The Ore-degree of is . We show that every graph with and is online -choosable. In addition, we prove an upper bound for online list-coloring triangle-free graphs: . Finally, we characterize Gallai trees as the connected graphs with no independent set incident to…
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