Extracting list colorings from large independent sets

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.