DP-Degree Colorable Hypergraphs

TL;DR

This paper extends the concept of DP-coloring from graphs to hypergraphs, characterizes DP-degree colorable hypergraphs, and establishes related structural and coloring bounds.

Contribution

It introduces DP-coloring for hypergraphs, provides a characterization of DP-degree colorable hypergraphs, and proves a Brooks' type theorem for hypergraph DP-chromatic number.

Findings

Characterization of DP-degree colorable hypergraphs

A Brooks' type theorem for hypergraph DP-chromatic number

Structural properties and bounds for DP-critical hypergraphs

Abstract

In order to solve a question on list coloring of planar graphs, Dvo\v{r}\'{a}k and Postle introduced the concept of so called DP-coloring, thereby extending the concept of list-coloring. DP-coloring was anaylized in detail by Bernshteyn, Kostochka, and Pron for graphs and multigraphs; they characterized DP-degree colorable multigraphs and deduced a Brooks' type result from this. The characterization of the corresponding 'bad' covers was later given by Kim and Ozeki. In this paper, the concept of DP-colorings is extended to hypergraphs having multiple (hyper-)edges. We characterize the DP-degree colorable hypergraphs and, furthermore, the corresponding 'bad' covers. This gives a Brooks' type result for the DP-chromatic number of a hypergraph. In the last part, we examine DP-critical graphs and establish some basic facts on their structure as well as a Gallai-type bound on the minimum number of edges.