# Generalized DP-Colorings of Graphs

## TL;DR

This paper extends the concept of DP-coloring to generalized graph colorings based on a property , establishing analogs of classical theorems and providing a broad framework for graph coloring with various constraints.

## Contribution

It introduces the -DP-chromatic number for graphs, generalizes fundamental coloring theorems, and extends a result on graph partitions into subgraphs with bounded degeneracy.

## Key findings

- Established -DP-chromatic number bounds for various graph classes
- Proved analogs of Brooks, Gallai, and Erd
51s-Rubin-Taylor theorems for -DP-coloring
- Generalized a 2000 result on graph partitions into subgraphs with bounded degeneracy

## Abstract

By a graph we mean a finite undirected graph having multiple edges but no loops. Given a graph property $\mathcal{P}$, a $\mathcal{P}$-coloring of a graph $G$ with color set $C$ is a mapping $\f:V(G)\to C$ such that for each color $c\in C$ the subgraph of $G$ induced by the color class $\varphi^{-1}(c)$ belongs to $\mathcal{P}$. The $\mathcal{P}$-chromatic number $\chi(G:\mathcal{P})$ of $G$ is the least number $k$ for which $G$ admits an $\mathcal{P}$-coloring with a set of $k$-colors. This coloring concept dates back to the late 1960s and is commonly known as generalized coloring. In the 1980s the $\mathcal{P}$-choice number $\chi_\ell(G:\mathcal{P})$ of $G$ was introduced and investigated by several authors. In 2018 \v{D}vor\'ak and Postle introduced the DP-chromatic number as a natural extension of the choice number. They also remarked that this concept applies to any graph property. This motivated us to investigate the $\mathcal{P}$-DP-chromatic number $\chi_{\rm DP}(G:\mathcal{P})$ of $G$. We have $\chi(G:\mathcal{P})\leq \chi_\ell(G:\mathcal{P})\leq \chi_{\rm DP}(G:\mathcal{P})$. In this paper we show that various fundamental coloring results, in particular, the theorems of Brooks, of Gallai, and of Erd\H{o}s, Rubin and Taylor, have counterparts for the $\mathcal{P}$-DP-chromatic number. Furthermore, we provide a generalization of a result from 2000 about partition of graphs into a fixed number of induced subgraphs with bounded variable degeneracy due to Borodin, Kostochka, and Toft.