Extracting list colorings from large independent sets

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 $xy$ in a graph $G$ is $\theta(xy) = d_G(x) + d_G(y)$. The Ore-degree of $G$ is $\theta(G) = \max_{xy\in E(G)}\theta(xy)$. We show that every graph with $\theta\ge18$ and $\omega\le\frac{\theta}{2}$ is online $\left\lfloor \frac{\theta}{2}\right\rfloor $-choosable. In addition, we prove an upper bound for online list-coloring triangle-free graphs: $\chi_{OL}\le\Delta+1-\lfloor\frac{1}{4}\lg(\Delta)\rfloor$. Finally, we characterize Gallai trees as the connected graphs $G$ with no independent set incident to at least $|G|$ edges.