Separating the online and offline DP-chromatic numbers

TL;DR

This paper demonstrates that the difference between online and offline DP-chromatic numbers of a graph can be made arbitrarily large by constructing specific graphs, highlighting a significant separation in these graph coloring parameters.

Contribution

The paper provides the first construction of graphs with arbitrarily large gaps between online and offline DP-chromatic numbers, answering an open question.

Findings

Constructed graphs with arbitrarily large online-offline DP-chromatic number gap

Confirmed the possibility of unbounded difference in DP-chromatic numbers

Extended understanding of the separation between online and offline graph coloring

Abstract

The DP-coloring problem is a generalization of the list-coloring problem in which the goal is to find an independent transversal in a certain topological cover of a graph $G$. In the online DP-coloring problem, the cover of $G$ is revealed one component at a time, and the independent transversal of the cover must be constructed in parts based on incomplete information. Kim, Kostochka, Li, and Zhu asked whether the chromatic numbers corresponding to these two graph coloring problems can have an arbitrarily large difference in a single graph. We answer this question in the affirmative by constructing graphs for which the gap between the online DP-chromatic number and the offline DP-chromatic number is arbitrarily large.